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%%% Multi-Linear Multiplication
% $\mlm_{k \in [r]}$ or $\mlm_{k = 1}^{r}$ (lower limit MUST be the first!)
% Save first argument as \arg@one
\def\mlm_#1{\def\arg@one{#1}\futurelet\next\mlm@i}
% Check for second argument
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% specialization for one or two arguments, both versions use saved first argument
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    {\operatorname*{\scalerel*[1.65em]{\times}{\bigotimes}}_{\arg@one}}%
    {\operatorname*{\scalerel*[1.65em]{\times}{\bigotimes}}_{\arg@one}}%
    {\operatorname*{\scalerel*[1.65em]{\times}{\bigotimes}}_{\arg@one}}%
}
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}

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    {\bigotimes_{\arg@one}^{#1}}%
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\begin{document}

%\tableofcontents

%------------------------------------------------------------------------------%
\section{Introduction}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Notation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We start by introducing the notation we use throughout the paper. Let $\ten{A} = (\ten{A}_{i_1, \ldots, i_r})\in\mathbb{R}^{q_1\times \ldots\times q_r}$ be an order\footnote{Also referred to as rank, therefore the variable name $r$, but this term is \emph{not} used as it leads to confusion with the rank as in ``the rank of a matrix''.} $r$ tensor, where $r\in\mathbb{N}$ is the number of modes or axes (dimensions) of $\ten{A}$ and $\ten{A}_{i_1,...,i_r} \in \mathbb{R}$ is its $(i_1, \ldots, i_r)$th entry. For example, a $p \times q$ matrix  $\mat{B}$ has two modes, the rows and columns.  For matrices $\mat{B}_k\in\mathbb{R}^{p_k\times q_k}$, $k\in[r] = \{1, 2, \ldots, r\}$, the \emph{multi-linear multiplication}, or \emph{Tucker Operator} \parencite{MultilinearOperators-Kolda2006}, is defined element wise as
\begin{displaymath}
    (\ten{A}\times\{\mat{B}_1, \ldots, \mat{B}_r\})_{j_1, \ldots, j_r} = \sum_{i_1, \ldots, i_r = 1}^{q_1, \ldots, q_r} \ten{A}_{i_1, \ldots, i_r}(\mat{B}_{1})_{j_1, i_1} \cdots (\mat{B}_{r})_{j_r, i_r}
\end{displaymath}
which results in an order $r$ tensor of dimension $p_1\times ...\times p_k$. This results in the \emph{$k$-mode product} between the tensor $\ten{A}$ with the matrix $\mat{B}_k$,
\begin{displaymath}
    \ten{A}\times_k\mat{B}_k = \ten{A}\times\{\mat{I}_{q_1}, \ldots, \mat{I}_{q_{k-1}}, \mat{B}_{k}, \mat{I}_{q_{k+1}}, \ldots, \mat{I}_{q_r}\}.
\end{displaymath}
Furthermore, the notation $\ten{A}\mlm_{k\in S}\mat{B}_k$ is short hand for the iterative application of the mode product for all indices in $S\subseteq[r]$. For example $\ten{A}\times_{k\in\{2, 5\}}\mat{B}_k = \ten{A}\times_2\mat{B}_2\times_5\mat{B}_5$. By only allowing $S$ to be a set, this notation is unambiguous, because the mode product commutes for different modes: $j\neq k\Rightarrow\ten{A}\times_j\mat{B}_j\times_k\mat{B}_k = \ten{A}\times_k\mat{B}_k\times_j\mat{B}_j$.

%Matrices and tensors can be \emph{vectorized} by the \emph{vectorization} operator $\vec$. 
The operator $\vec$ maps an array to a vector. For example,  $\vec(\mat{B})$ stands for the $pq \times 1$ vector of the $p \times q$ matrix $\mat{B}$ resulting from stacking the columns of $\mat{B}$ one after the other.  For a tensor $\ten{A}= (a_{i_1,\ldots,i_r})$ of order $r$ and dimensions $q_1, \ldots, q_r$, $\vec(\ten{A})$ is the $q_1 q_2 \ldots q_r \times 1$ vector with the elements of $\ten{A}$ stacked one after the other in the specified order $r$ then $r-1$, and so on. For example, if $\ten{A}$ is 3-dimensional array, $\vec(\ten{A})=(\vec(\ten{A}(:,:,1))^T,\vec(\ten{A}(:,:,2)^T,\ldots,\vec(\ten{A}(:,:,q_r)^T)^T$. We use the notation $\ten{A}\equiv \ten{B}$ for objects $\ten{A}, \ten{B}$ of any shape if and only if $\vec(\ten{A}) = \vec(\ten{B})$.

The \emph{inner product} between two tensors of the same order and dimensions is
\begin{displaymath}
    \langle\ten{A}, \ten{B}\rangle = \sum_{i_1, \ldots, i_r} \ten{A}_{i_1, \ldots, i_r}\ten{B}_{i_1, \ldots, i_r}
\end{displaymath}
that leads to the definition of the \emph{Frobenius Norm} for tensors, $\|\ten{A}\|_F = \sqrt{\langle\ten{A}, \ten{A}\rangle}$ and is the straightforward  extension of the Frobenius norm for matrices and vectors. %are also used for matrices while for a vector $\mat{a}$ the \emph{2 norm} is $\|\mat{a}\|_2 = \sqrt{\langle\mat{a}, \mat{a}\rangle}$.
The \emph{outer product} between two tensors $\ten{A}$ of dimensions $q_1, \ldots, q_r$ and $\ten{B}$ of dimensions $p_1, \ldots, p_l$ is a tensor $\ten{A}\circ\ten{B}$ of order $r + l$ and dimensions $q_1, \ldots, q_r, p_1, \ldots, p_l$ such that
\begin{displaymath}
    \ten{A}\circ\ten{B} \equiv (\vec\ten{A})\t{(\vec{\ten{B}})}.
\end{displaymath}
Let $\K : \mathbb{R}^{q_1, ..., q_{2 r}}\to\mathbb{R}^{q_1 q_{r + 1}, ..., q_r q_{2 r}}$ be defined element wise with indices $1\leq i_j + 1\leq q_j q_{r + j}$ for $j = 1, ..., r$ as
\begin{align*}
    \K(\ten{A})_{i_1 + 1, ..., i_r + 1} &= \ten{A}_{\lfloor i_1 / q_{r + 1}\rfloor + 1, ..., \lfloor i_r / q_{2 r} \rfloor + 1, (i_1\operatorname{mod}q_{r + 1}) + 1, ..., (i_r\operatorname{mod}q_{2 r}) + 1}
\end{align*}
where $\lfloor\,.\,\rfloor$ is the floor operation and $a\operatorname{mod}b$ is the integer divition remainder of $a / b$. The mapping $\K$ is a linear operation and maps an order $2 r$ tensor to an order $r$ tensor by reshaping and permuting its elements. This operation allows to define a generalization of the \emph{Kronecker product} which we define as $\ten{A}\otimes\ten{B} = \K(\ten{A}\circ\ten{B})$.

For tensors of order at least $2$, the \emph{flattening} (or \emph{unfolding} or \emph{matricization}) is a reshaping of the tensor into a matrix along a particular mode. For a tensor $\ten{A}$ of order $r$ and dimensions $q_1, \ldots, q_r$, the $k$-mode unfolding $\ten{A}_{(k)}$ is a $q_k\times \prod_{l=1, l\neq k}q_l$ matrix with %For the tensor $\ten{A} = (a_{i_1,\ldots,i_r})\in\mathbb{R}^{q_1, \ldots, q_r}$ the 
elements %of the $k$ unfolded tensor $\ten{A}_{(k)}$ are
\begin{displaymath}
    (\ten{A}_{(k)})_{i_k, j} = \ten{A}_{i_1, \ldots, i_r}\quad\text{ with }\quad j = 1 + \sum_{\substack{l = 1\\l \neq k}}^r (i_l - 1) \prod_{\substack{m = 1\\m\neq k}}^{l - 1}q_m.
\end{displaymath}
% The rank of a tensor $\ten{A}$ of dimensions $q_1\times ...\times q_r$ is vector-valued; that is, $\rank(\ten{A}) = (a_1, \ldots, a_r)\in[q_1]\times...\times[q_r]$, where $a_k = \rank(\ten{A}_{(k)})$ is the usual matrix rank of the $k$ unfolded tensor.

The gradient of a function $\ten{F}(\ten{X})$ of any shape, univariate, multivariate or tensor valued, with argument $\ten{X}$ of any shape is defined as
\begin{displaymath}
    \nabla_{\ten{X}}\ten{F} = \frac{\partial\t{(\vec\ten{F}(\ten{X}))}}{\partial(\vec\ten{X})}
\end{displaymath}
which is a matrix of dimension $p\times q$ where the vectorized quantities $\vec{\ten{X}}\in\mathbb{R}^p$ and $\vec\ten{F}(\ten{X})\in\mathbb{R}^q$. This is consistent with the gradient of a real-valued function $f(\mat{x})$ where $\mat{x}\in\mathbb{R}^p$ as $\nabla_{\mat{x}}f\in\mathbb{R}^{p\times 1}$. \todo{Maybe reference magnus and abadir, magnus and neudecker?!}



\todo{$\vech\ten{A}$ the half vectorization! Define via the vector containing the tensor elements with indices in \emph{reflected lexicographic order}? Also don't forget to figure out how to (if even) to define the half vectorization of a tensor (with all modes of the same dimensions)}

\todo{$\sym{\ten{A}}$ tensor needed?!}

\todo{I think the following examples are a good idea for the appendix.}

\begin{example}[Vectorization]\label{ex:vectorization}
    Given a matrix
    \begin{displaymath}
        \mat{A} = \begin{pmatrix}
            1 & 4 & 7 \\
            2 & 5 & 8 \\
            3 & 6 & 9
        \end{pmatrix}
    \end{displaymath}
    its vectorization is $\vec{\mat{A}} = \t{(1, 2, 3, 4, 5, 6, 7, 8, 9)}$ and its half vectorization $\vech{\mat{A}} = \t{(1, 2, 3, 5, 6, 9)}$. Let a $\ten{A}$ be a tensor of dimension $3\times 3\times 3$ given by
    \begin{displaymath}
        \ten{A}_{:,:,1} = \begin{pmatrix}
            1 & 4 & 7 \\
            2 & 5 & 8 \\
            3 & 6 & 9
        \end{pmatrix},
        \qquad
        \ten{A}_{:,:,2} = \begin{pmatrix}
            10 & 13 & 16 \\
            11 & 14 & 17 \\
            12 & 15 & 18
        \end{pmatrix},
        \qquad
        \ten{A}_{:,:,3} = \begin{pmatrix}
            19 & 22 & 25 \\
            20 & 23 & 26 \\
            21 & 24 & 27
        \end{pmatrix}.
    \end{displaymath}
    Then the vectorization of $\ten{A}$ is given by
    \begin{displaymath}
        \vec{\ten{A}} = \t{(1, 2, 3, 4, ..., 26, 27)}\in\mathbb{R}^{27}
    \end{displaymath}
    while the half vectorization is
    \begin{displaymath}
        \vech{\ten{A}} = \t{(1, 2, 3, 5, 6, 9, 11, 12, 15, 21)}\in\mathbb{R}^{10}.
    \end{displaymath}
\end{example}

\begin{example}[Matricization]
    Let $\ten{A}$ be the $3\times 4\times 2$ tensor given by
    \begin{displaymath}
        \ten{A}_{:,:,1} = \begin{pmatrix}
            1 & 4 & 7 & 10 \\
            2 & 5 & 8 & 11 \\
            3 & 6 & 9 & 12
        \end{pmatrix},
        \ten{A}_{:,:,2} = \begin{pmatrix}
            13 & 16 & 19 & 22 \\
            14 & 17 & 20 & 23 \\
            15 & 18 & 21 & 24
        \end{pmatrix}.
    \end{displaymath}
    Its matricizations are then
    \begin{gather*}
        \ten{A}_{(1)} = \begin{pmatrix}
            1 &  4 &  7 & 10 & 13 & 16 & 19 & 22 \\
            2 &  5 &  8 & 11 & 14 & 17 & 20 & 23 \\
            3 &  6 &  9 & 12 & 15 & 18 & 21 & 24
        \end{pmatrix},
        \qquad
        \ten{A}_{(2)} = \begin{pmatrix}
             1 &  2 &  3 & 13 & 14 & 15 \\
             4 &  5 &  6 & 16 & 17 & 18 \\
             7 &  8 &  9 & 19 & 20 & 21 \\
            10 & 11 & 12 & 22 & 23 & 24
        \end{pmatrix}, \\
        \ten{A}_{(3)} = \begin{pmatrix}
             1 &  2 &  3 &  4 &  5 &  6 &  7 &  8 &  9 &  10 &  11 &  12 \\
            13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 &  22 &  23 &  24
        \end{pmatrix}.
    \end{gather*}
\end{example}

% \begin{example}[Symmetrization]
%     Let $\ten{A}$ be the $3\times 3\times 3$ tensor from \cref{ex:vectorization}, then the symmetrization of $\ten{A}$ is
%     \begin{align*}
%         (\sym{\ten{A}})_{:,:,1} &= \frac{1}{6}\begin{pmatrix}
%              6 & 32 &  58 \\
%             32 & 58 &  84 \\
%             58 & 84 & 110
%         \end{pmatrix}, \\
%         (\sym{\ten{A}})_{:,:,2} &= \frac{1}{6}\begin{pmatrix}
%             32 &  58 &  84 \\
%             58 &  84 & 110 \\
%             84 & 110 & 136
%         \end{pmatrix}, \\
%         (\sym{\ten{A}})_{:,:,3} &= \frac{1}{6}\begin{pmatrix}
%              58 &  84 & 110 \\
%              84 & 110 & 136 \\
%             110 & 136 & 162
%         \end{pmatrix}.
%     \end{align*}
% \end{example}

% \begin{example}[Half Vectorization]
%     The half vectorization of a square matrix
%     \begin{displaymath}
%         \mat{A} = \begin{pmatrix}
%             1 & 4 & 7 \\
%             2 & 5 & 8 \\
%             3 & 6 & 9
%         \end{pmatrix}
%     \end{displaymath}
%     is
%     \begin{displaymath}
%         \vech{\mat{A}} = (1, 2, 3, 5, 6, 9).
%     \end{displaymath}
% \end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Problem Formulation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Our goal is to identify the cumulative distribution function (cdf) $F$ of $Y\mid \ten{X}$, where $\ten{X}$ is assumed to admit $r$-tensor structure of dimension $p_1\times ... \times p_r$ with continuous or discrete entries and there is no constraint in the form of $Y$. The predictor $\ten{X}$ is a complex object; to simplify the problem we assume their exists a function $\ten{R}:\ten{X}\mapsto \ten{R}(\ten{X})$ such that $\ten{R}(\ten{X})$ is tensor valued of lower dimension so that
\begin{displaymath}
    F(Y\mid \ten{X}) = F(Y\mid \ten{R}(\ten{X})).
\end{displaymath}
Since $\ten{R}(\ten{X})$ replaces the predictors without any effect in the conditional cdf of $Y\mid \ten{X}$, it is a \emph{sufficient reduction} for the regression $Y\mid\ten{X}$. We assume the tensor valued $\ten{R}(\ten{X})$ has dimension  $q_1\times...\times q_r$ with $q_j\leq p_j$, $j = 1, ..., r$, which represents a dimension reduction along every mode of $\ten{X}$. This formulation is flexible as it allows, for example, to select ``important'' modes by reducing ``unimportant'' modes to be $1$ dimensional.

To find such a reduction $\ten{R}$, we leverage the equivalence pointed out in \textcite{FisherLectures-Cook2007},
\begin{equation}\label{eq:inverse-regression-sdr}
    Y\mid\ten{X} \sim Y\mid \ten{R}(\ten{X})
        \quad\Longleftrightarrow\quad
    \ten{X}\mid(Y, \ten{R}(\ten{X})) \sim \ten{X}\mid\ten{R}(\ten{X}),
\end{equation}
which means that a \textit{sufficient statistic} $\ten{R}(\ten{X})$ for $Y$ in the inverse regression $\ten{X}\mid Y$, where $Y$ is considered as a parameter indexing the model, is also a sufficient reduction for $\ten{X}$ in the forward regression $Y\mid\ten{X}$. The equivalent inverse regression in \eqref{eq:inverse-regression-sdr} provides exhaustive characterization of $\ten{R}(\ten{X})$. 

The usual tool to identify sufficient statistics is the factorization theorem that requires a distributional model. Here we assume the distribution of $\ten{X}\mid Y$ belongs to  the \emph{quadratic exponential family} in order to (a) simplify modeling and (b) keep estimation feasible. An important feature of the \emph{quadratic exponential family} is that its members are characterized by their first two moments. Specifically, we assume that $\ten{X}\mid Y$ is a full rank quadratic exponential family with density
\begin{align}
f_{\mat{\eta}_y}(\ten{X}\mid Y = y)
    &= h(\ten{X})\exp(\t{\mat{\eta}_y}\mat{t}(\ten{X}) - b(\mat{\eta}_y)) \nonumber \\
    &= h(\ten{X})\exp(\langle \mat{t}_1(\ten{X}), \mat{\eta}_{1y} \rangle + \langle \mat{t}_2(\ten{X}), \mat{\eta}_{2y} \rangle - b(\mat{\eta}_{y})) \label{eq:quad-density}
\end{align}
where $\mat{t}_1(\ten{X})=\vec \ten{X}$ and $\mat{t}_2(\ten{X})$ is linear in $\ten{X}\circ\ten{X}$. The dependence of $\ten{X}$ on $Y$ is fully captured in the natural parameter $\mat{\eta}_y$. The function $h$ is non-negative real-valued. For $b$ we assume it is at least twice continuously differentiable and structly convex.
Distributions within the quadratic exponential family include the \emph{tensor normal} \todo{cite, if can be found} and \emph{tensor Ising model} \todo{cite} (a generalization of the (inverse) Ising model which is multi-variate Bernoulli with up to second order interactions) and mixtures of these two.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Generalized Multi-Linear Model}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In model \eqref{eq:quad-density}, the relationship of $\ten{X}$ and $Y$ is absorbed in $\mat{\eta}_y$, and $\mat{t}(\ten{X})$ is the minimal sufficient statistic for the \textit{pseudo}-parameter\footnote{$\mat{\eta}_y$ is a function of the response $Y$, thus it is not a parameter in the formal statistical sense. It is considered as a parameter when using the equivalence in \eqref{eq:inverse-regression-sdr} and view  $Y$ as a parameter as a device to derive the sufficient reduction from the inverse regression.} $\mat{\eta}_y = (\mat{\eta}_{1y}, \mat{\eta}_{2y})$ with
\begin{align}\label{eq:t-stat}
    \mat{t}(\ten{X}) &= (\mat{t}_1(\ten{X}),\mat{t}_2(\ten{X}))=(\vec{\ten{X}}, \mat{T}_2\vech((\vec\ten{X})\t{(\vec\ten{X})})),
\end{align}
where the $d\times p(p + 1) / 2$ dimensional matrix $\mat{T}_2$ with $p = \prod_{i = 1}^r p_i$ ensures that $\mat{\eta}_{2y}$ is of minimal dimension $d$. The matrix $\mat{T}_2$ is of full rank $d$ and is unique to specific members of the quadratic exponential family.

We can reexpress the exponent in \eqref{eq:quad-density} as
\begin{align*}
    \t{\mat{\eta}_y} \mat{t}(\ten{X})
        &= \langle \vec \ten{X}, \mat{\eta}_{1y} \rangle + \langle \mat{T}_2\vech(\ten{X}\circ\ten{X}), \mat{\eta}_{2y} \rangle = \langle \vec \ten{X}, \mat{\eta}_{1y} \rangle + \langle \vec(\ten{X}\circ\ten{X}), \t{(\mat{T}_2\pinv{\mat{D}_p})}\mat{\eta}_{2y} \rangle
\end{align*}
where $\mat{D}_p$ is the \emph{duplication matrix} from \textcite[Ch.~11]{MatrixAlgebra-AbadirMagnus2005}, defined so that $\mat{D}_p\vech \mat{A} = \vec \mat{A}$ for every symmetric $p\times p$ matrix $\mat{A}$, and $\pinv{\mat{D}_p}$ is its Moore-Penrose pseudo inverse. The first natural parameter component, $\mat{\eta}_{1y}$, captures the first order, and $\mat{\eta}_{2y}$, the second order relationship of $Y$ and $\ten{X}$. The quadratic exponential density of $\ten{X} \mid Y$ can then be expressed as
\begin{equation}\label{eq:quadratic-exp-fam}
    f_{\eta_y}(\ten{X}\mid Y = y) = h(\ten{X})\exp(\langle \vec \ten{X}, \mat{\eta}_{1y} \rangle + \langle \vec(\ten{X}\circ\ten{X}), \t{(\mat{T}_2\pinv{\mat{D}_p})}\mat{\eta}_{2y} \rangle - b(\mat{\eta}_y))
\end{equation}
The exponential family in \eqref{eq:quadratic-exp-fam} is easily generalizable to any order. This, though, would result in the number of parameters becoming prohibitive to estimate. This is also the reason why we opted for the second order exponential family in our formulation.

By the equivalence in \eqref{eq:inverse-regression-sdr}, in order to find the sufficient reduction $\ten{R}(\ten{X})$ we need to infer $\mat{\eta}_{1y}$, and $\mat{\eta}_{2y}$. This is reminiscent of generalized linear modeling, which we extend to a multi-linear formulation next.
Suppose $\ten{F}_y$ is a known mapping of $y$ with zero expectation $\E_Y\ten{F}_Y = 0$. We assume the dependence of $\ten{X}$ and $Y$ is reflected only in the first parameter and let
\begin{align}
    \mat{\eta}_{1y} &= \vec{\overline{\ten{\eta}}} + \mat{B}\vec\ten{F}_y, \label{eq:eta1-manifold} \\
    \mat{\eta}_{2}  &= \t{(\pinv{(\mat{T}_2\pinv{\mat{D}_p})})}\vec(c\,\mat{\Omega}), \label{eq:eta2-manifold}
\end{align}
where $\overline{\ten{\eta}}\in\mathbb{R}^{p_1\times\ldots\times p_r}$, $\mat{\Omega} \in \mathbb{R}^{p \times p}$ is positive definite with $p = \prod_{j = 1}^{r} p_j$, and $c\in\mathbb{R}$ is a known constant determined by the distribution to ease modeling. That is, we assume that only $\mat{\eta}_{1y}$ depends on $Y$ through $\mat{B}$. The second parameter $\mat{\eta}_2$ captures the second order interaction structure of $\ten{X}$, which we assume not to depend on the response $Y$. In order to relate individual modes of $\ten{X}$ to the response, allowing flexibility in modeling, we assume $\ten{F}_y$ takes values in $\mathbb{R}^{q_1\times ...\times q_r}$; that is, $\ten{F}_y$ is a tensor valued independent variable in the inverse regression setting. This, in turn, leads to imposing corresponding tensor structure to the regression parameter $\mat{B}$. Thus, \eqref{eq:eta1-manifold} becomes
\begin{align}
    \mat{\eta}_{1y} &=
    \vec\biggl(\overline{\ten{\eta}} + \ten{F}_y\mlm_{j = 1}^{r}\mat{\beta}_j\biggr), \label{eq:eta1}
\end{align}
where $\mat{B} = \bigotimes_{j = r}^{1}\mat{\beta}_j$ and the component matrices $\mat{\beta}_j\in\mathbb{R}^{p_j\times q_j}$ are of known rank for $j = 1, \ldots, r$. Given the high potential value of $p$, we further assume that
\begin{align}
    \t{(\mat{T}_2\pinv{\mat{D}_p})}\mat{\eta}_{2y}= \t{(\mat{T}_2\pinv{\mat{D}_p})}\mat{\eta}_{2} &= \vec\biggl(c\bigotimes_{j = r}^{1}\mat{\Omega}_j\biggr). \label{eq:eta2}
\end{align}
where $\mat{\Omega}_j\in\mathbb{R}^{p_j\times p_j}$ are symmetric positive definite matrices for $j = 1, \ldots, r$. That is, we require $\mat{\Omega} = \bigotimes_{j = r}^{1}\mat{\Omega}_j$, which substantially reduces the number of parameters to estimate in $\mat{\Omega}$. The assumption that the $\mat{\Omega}_j$'s be positive definite is possible due to the constant $c$.
Equation \eqref{eq:eta2} is underdetermined since $\t{(\mat{T}_2\pinv{\mat{D}_p})}$ has full column rank $d < p^2$ (with a non-strict inequality if $\ten{X}$ is univariate) but $\mat{\eta}_2$ is uniquely determined given any $\mat{\Omega}$ as $\t{(\pinv{(\mat{T}_2\pinv{\mat{D}_p})})}$ has full row rank. We let $\mat{\xi} = (\vec{\overline{\ten{\eta}}}, \vec{\mat{B}}, \vech{\mat{\Omega}})$ be a $p(p + 2 q + 3) / 2$-parameter vector and $\mat{\theta} = (\vec{\overline{\ten{\eta}}}, \vec{\mat{B}}, \vech{\mat{\Omega}})$ be the constrained parameter vector, where $\mat{B}=\bigotimes_{j = r}^{1}\mat{\beta}_j$ and $\mat{\Omega} = \bigotimes_{j = r}^{1}\mat{\Omega}_j$. We also let $\Xi$ and $\Theta$ denote the unconstrained and constrained parameter spaces, with $\mat{\xi}$ and $\mat{\theta}$  varying in $\Xi$ and $\Theta$, respectively. The parameter space $\Xi$ is an open subset of $\mathbb{R}^{p(p + 2 q + 3) / 2}$ so that \eqref{eq:quadratic-exp-fam} is a proper density. Later, we relax the assumptions for $\mat{\beta}_k$ and $\mat{\Omega}_k$ as a consequence of \cref{thm:param-manifold} in \cref{sec:kron-manifolds}.

In a classical \emph{generalized linear model} (GLM), the link function connecting the natural parameters to the expectation of the sufficient statistic $\mat{\eta}_y = \mat{g}(\E[\mat{t}(\ten{X}) \mid Y = y])$ is invertible. Such a link may not exist in our setting, but for our purpose what we call the ``inverse'' link suffices. The ``inverse'' link $\widetilde{\mat{g}}$ exists as the natural parameters fully describe the distribution. As in the non-minimal formulation \eqref{eq:quadratic-exp-fam}, we define the ``inverse'' link through its tensor valued components
\begin{align}
    \ten{g}_1(\mat{\eta}_y) &= \E[\ten{X} \mid Y = y], \label{eq:inv-link1}\\
    \ten{g}_2(\mat{\eta}_y) &= \E[\ten{X}\circ\ten{X} \mid Y = y] \label{eq:inv-link2}
\end{align}
as $\widetilde{\mat{g}}(\mat{\eta}_y) = (\vec\ten{g}_1(\mat{\eta}_y), \vec\ten{g}_2(\mat{\eta}_y))$.

Under the quadratic exponential family model \eqref{eq:quadratic-exp-fam},  a sufficient reduction for the regression of $Y$ on $\ten{X}$ is given in \cref{thm:sdr}.
\begin{theorem}[SDR]\label{thm:sdr}
    A sufficient reduction for the regression $Y\mid \ten{X}$ under the quadratic exponential family inverse regression model \eqref{eq:quadratic-exp-fam} with natural parameters \eqref{eq:eta1} and \eqref{eq:eta2} is given by
    \begin{align}\label{eq:sdr}
        \ten{R}(\ten{X})
            % &= (\ten{X} - \E\ten{X})\times\{ \t{\mat{\beta}_1}, \ldots, \t{\mat{\beta}_r} \}.
            &= (\ten{X} - \E\ten{X})\mlm_{k = 1}^{r}\t{\mat{\beta}_j}.
    \end{align}
    The reduction \eqref{eq:sdr} is minimal if $\mat{\beta}_j$ are full rank for all $j=1,\ldots,r$.
\end{theorem}

The reduction in vectorized form is $\vec\ten{R}(\ten{X})=\t{\mat{B}}\vec(\ten{X} - \E\ten{X})$, where $\mat{B} = \bigotimes_{k = r}^{1}\mat{\beta}_k$ with $\Span(\mat{B}) = \Span(\{\mat{\eta}_{1y} - \E_{Y}\mat{\eta}_{1Y} : y\in\mathcal{S}_Y\})$, using $\mathcal{S}_Y$ to denote the set of values of the random variable $Y$.

\cref{thm:sdr} obtains that the \emph{sufficient reduction} $\ten{R}(\ten{X})$ reduces $\ten{X}$ along each dimension linearly. The graph in \cref{fig:SDRvisual} is a visual depiction of the sufficient reduction.

\begin{figure}
    \centering
    \begin{scaletikzpicturetowidth}{0.5\textwidth}
        \input{images/reduction.tex}
    \end{scaletikzpicturetowidth}
    \caption{\label{fig:SDRvisual}Visual depiction of the sufficient reduction in \cref{thm:sdr}.}
\end{figure}


\begin{example}[Vector valued $\mat{x}$ ($r = 1$)]\label{ex:vector-dist}
    Given vector valued predictor $\mat{X}\in\mathbb{R}^p$, the tensor order is $r = 1$, then the collection of parameters is $\mat{\theta} = (\overline{\mat{\eta}}, \mat{\beta}, \mat{\Omega})$ with $\overline{\mat{\eta}}\in\mathbb{R}^p$, $\mat{\beta}\in\StiefelNonCompact{p}{q}$ and $\mat{\Omega}\in\SymPosDefMat{p}$ where $\mat{f}_y\in\mathbb{R}^q$ are known functions of the response $Y$. The conditional density of $\mat{X}\mid Y = y$ is given by
    \begin{align*}
        f_{\theta}(\mat{x}\mid Y = y)
            &= h(\mat{x})\exp(\langle\mat{x}, \mat{\eta}_{1y}(\mat{\theta})\rangle + \langle\vec(\mat{x}\circ\mat{x}), \mat{\eta}_2(\mat{\theta})\rangle - b(\mat{\eta}_y(\mat{\theta}))) \\
            % &= h(\mat{x})\exp(\t{\mat{\eta}_{1y}(\theta)}\mat{x} + \t{\vec(\mat{x}\circ\mat{x})}\mat{\eta}_2(\mat{\theta}) - b(\mat{\eta}_y(\mat{\theta}))) \\
            &= h(\mat{x})\exp(\t{(\overline{\mat{\eta}} + \mat{\beta}\mat{f}_y)}\mat{x} + c\,\t{\mat{x}}\mat{\Omega}\,\mat{x} - b(\mat{\eta}_y(\mat{\theta}))).
    \end{align*}
    using the relation of $\mat{\theta}$ to the natural parameters given by $\mat{\eta}_{1y}(\mat{\theta}) = \overline{\mat{\eta}} + \mat{\beta}\mat{f}_y$ and $\mat{\eta}_2(\theta) = c\,\mat{\Omega}$.
    % where the number of unknown parameters is $p + \dim(\StiefelNonCompact{p}{q}) + \dim(\SymPosDefMat{p}) = p\frac{p + 2 q + 3}{2}$.
\end{example}

\begin{example}[Matrix valued $\mat{X}$ ($r = 2$)]
    Assuming $\mat{X}$ to be matrix valued, that is $r = 2$, $\mat{\theta} = (\overline{\mat{\eta}}, \mat{\beta}_1, \mat{\beta}_2, \mat{\Omega}_1, \mat{\Omega}_2)$, where the intercept term $\overline{\mat{\eta}}\in\mathbb{R}^{p_1\times p_2}$ is now matrix valued. Similar to \cref{ex:vector-dist} with $\mat{F}_y\in\mathbb{R}^{q_1\times q_2}$ being matrix valued, the conditional density of $\mat{X}\mid Y = y$ reads
    \begin{align*}
        f_{\mat{\theta}}(\mat{X}\mid Y = y)
            &= h(\mat{X})\exp(\langle\vec{\mat{X}}, \mat{\eta}_{1y}(\mat{\theta})\rangle + \langle\vec(\mat{X}\circ\mat{X}), \mat{\eta}_2(\mat{\theta})\rangle - b(\mat{\eta}_y(\mat{\theta}))) \\
            &= h(\mat{X})\exp(\tr((\overline{\mat{\eta}} + \mat{\beta}_1\mat{F}_y\t{\mat{\beta}_2})\t{\mat{X}}) + c \tr(\mat{\Omega}_1\mat{X}\mat{\Omega}_2\t{\mat{X}}) - b(\mat{\eta}_y(\mat{\theta}))).
    \end{align*}
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Maximum Likelihood Estimation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Suppose $(\ten{X}_i, Y_i)$ are independently and identically distributed with joint cdf $F(\ten{X}, Y)$, for $i = 1, \ldots, n$. The empirical log-likelihood function of \eqref{eq:quadratic-exp-fam} under \eqref{eq:eta1} and \eqref{eq:eta2}, ignoring terms not depending on the parameters, is
\begin{equation}\label{eq:log-likelihood}
    l_n(\mat{\theta}) = \frac{1}{n}\sum_{i = 1}^n \biggl(\Bigl\langle\overline{\ten{\eta}} + \ten{F}_{y_i}\mlm_{k = 1}^{r}\mat{\beta}_k, \ten{X}_i \Bigr\rangle + c\Bigl\langle\ten{X}_i\mlm_{k = 1}^{r}\mat{\Omega}_k, \ten{X}_i \Bigr\rangle - b(\mat{\eta}_{y_i})\biggr).
\end{equation}
The maximum likelihood estimate of $\mat{\theta}_0$ is the solution to the optimization problem
\begin{equation}\label{eq:mle}
    \hat{\mat{\theta}}_n = \argmax_{\mat{\theta}\in\Theta}l_n(\mat{\theta})
\end{equation}
with $\hat{\mat{\theta}}_n = (\vec\widehat{\overline{\ten{\eta}}}, \vec\widehat{\mat{B}}, \vech\widetilde{\mat{\Omega}})$ where $\widehat{\mat{B}} = \bigkron_{k = r}^{1}\widehat{\mat{\beta}}_k$ and $\widehat{\mat{\Omega}} = \bigkron_{k = r}^{1}\widehat{\mat{\Omega}}_k$.

A straightforward and general method for parameter estimation is \emph{gradient descent}. To apply gradient based optimization, we compute the gradients of $l_n$ in \cref{thm:grad}.

\begin{theorem}\label{thm:grad}
    For $n$ i.i.d. observations $(\ten{X}_i, y_i), i = 1, ..., n$ the log-likelihood is of the form in \eqref{eq:log-likelihood} with $\mat{\theta}$ being the collection of all GMLM parameters $\overline{\ten{\eta}}$, ${\mat{B}} = \bigkron_{k = r}^{1}{\mat{\beta}}_k$ and ${\mat{\Omega}} = \bigkron_{k = r}^{1}{\mat{\Omega}}_k$ for $k = 1, ..., r$. Let $\ten{G}_2(\mat{\eta}_y)$ be a tensor of dimensions $p_1, \ldots, p_r$ such that
    \begin{displaymath}
        \vec{\ten{G}_2(\mat{\eta}_y)} = \pinv{(\mat{T}_2\pinv{\mat{D}_p})}\mat{T}_2\pinv{\mat{D}_p}\vec{\ten{g}_2(\mat{\eta}_y)}.
    \end{displaymath}
    Then, the partial gradients with respect to $\overline{\ten{\eta}}, \mat{\beta}_1, \ldots, \mat{\beta}_r, \mat{\Omega}_1, \ldots, \mat{\Omega}_r$ are given by
    \begin{align*}
        \nabla_{\overline{\ten{\eta}}}l_n &= \vec\frac{1}{n}\sum_{i = 1}^n (\ten{X}_i - \ten{g}_1(\mat{\eta}_{y_i})), \\
        \nabla_{\mat{\beta}_j}l_n &= \vec\frac{1}{n}\sum_{i = 1}^n (\ten{X}_i - \ten{g}_1(\mat{\eta}_{y_i}))_{(j)}\t{\Big(\ten{F}_{y_i}\mlm_{k\in[r]\backslash j}\mat{\beta}_k\Big)_{(j)}}, \\
        \nabla_{\mat{\Omega}_j}l_n &= \vec\frac{c}{n}\sum_{i = 1}^n (\ten{X}_i\otimes\ten{X}_i - \K(\ten{G}_2(\mat{\eta}_{y_i})))\mlm_{k\in[r]\backslash j}\t{(\vec{\mat{\Omega}_k})}
    \end{align*}
    which obtains $\nabla l_n = (\nabla_{\overline{\ten{\eta}}}l_n, \nabla_{\mat{\beta}_1}l_n, \ldots, \nabla_{\mat{\beta}_r}l_n, \nabla_{\mat{\Omega}_1}l_n, \ldots, \nabla_{\mat{\Omega}_r}l_n)$.
If $\mat{T}_2$ is the identity matrix $\mat{I}_{p(p + 1) / 2}$, then $\ten{G}_2(\mat{\eta}_y) = \ten{g}_2(\mat{\eta}_y)$.
\end{theorem}

Although the general case of any GMLM model can be fitted via gradient descent using \cref{thm:grad}, this may be very inefficient. In \cref{thm:grad}, $\mat{T}_2$ can be used to introduce flexible second moment structures. For example, it allows modeling effects differently for predictor components, as described in \cref{sec:ising_estimation} after Eqn. \eqref{eq:ising-cond-prob}. In the remainder, we focus on $\mat{T}_2$'s that are identity matrices. This approach simplifies the estimation algorithm and the speed of the numerical calculation in the case of tensor normal predictors. In the case of the tensor normal distribution, an iterative cyclic updating scheme is derived in \cref{sec:tensor_normal_estimation}, which has much faster convergence, is stable and does not require hyper parameters, as will be discussed later. On the other hand, the Ising model does not allow such a scheme. There we need to use a gradient based method, which is the subject of \cref{sec:ising_estimation}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Tensor Normal}\label{sec:tensor_normal_estimation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Suppose $\ten{X}\mid Y = y$ follows a tensor normal distribution  with mean $\ten{\mu}_y$ and covariance $\mat{\Sigma} = \bigkron_{k = r}^{1}\mat{\Sigma}_k$. We assume the distribution is non-degenerate which means that the covariances $\mat{\Sigma}_k$ are symmetric positive definite matrices. Its density is given by
\begin{displaymath}
    f_{\mat{\theta}}(\ten{X}\mid Y = y) = (2\pi)^{-p / 2}\prod_{k = 1}^{r}\det(\mat{\Sigma}_k)^{-p / 2 p_k}\exp\left( -\frac{1}{2}\left\langle\ten{X} - \ten{\mu}_y, (\ten{X} - \ten{\mu}_y)\mlm_{k = 1}^{r}\mat{\Sigma}_k^{-1} \right\rangle \right).
\end{displaymath}
For the sake of simplicity and w.l.o.g., we assume $\ten{X}$ has 0 marginal expectation; i.e., $\E\ten{X} = 0$. Rewriting this in the quadratic exponential family form \eqref{eq:quadratic-exp-fam}, determines the scaling constant $c = -1/2$. The relation to the GMLM parameters $\mat{\beta}_k$ and $\mat{\Omega}_k$, for $k = 1, \ldots, r$ is
\begin{equation}\label{eq:tnormal_cond_params}
    \ten{\mu}_y = \ten{F}_y\mlm_{k = 1}^{r}\mat{\Omega}_k^{-1}\mat{\beta}_k, \qquad
    \mat{\Omega}_k = \mat{\Sigma}_k^{-1},
\end{equation}
where we used that $\overline{\ten{\eta}} = 0$ due to $0 = \E\ten{X} = \E\E[\ten{X}\mid Y] = \E\ten{\mu}_Y$ in combination with $\E\ten{F}_Y = 0$. Additionally, all the $\mat{\Omega}_k$'s are symmetric positive definite, because the $\mat{\Sigma}_k$'s are. This lead to another simplification since then $\mat{T}_2$ in \eqref{eq:t-stat} equals the identity. This also means that the gradients of the log-likelihood $l_n$ in \cref{thm:grad} are simpler. We obtain
\begin{displaymath}
    \ten{g}_1(\mat{\eta}_y) = \E[\ten{X}\mid Y = y] = \ten{\mu}_y, \qquad
    \ten{G}_2(\mat{\eta}_y) = \ten{g}_2(\mat{\eta}_y) = \E[\ten{X}\circ\ten{X}\mid Y = y] \equiv \bigkron_{k = r}^1\mat{\Sigma}_k + (\vec{\ten{\mu}}_y)\t{(\vec{\ten{\mu}}_y)}.
\end{displaymath}
In practice, we assume we have a random sample of $n$ observations $(\ten{X}_i, \ten{F}_{y_i})$ from the joint distribution. We start the estimation process by demeaning them. Then, only the reduction matrices $\mat{\beta}_k$ and the scatter matrices $\mat{\Omega}_k$ need to be estimated. To solve the optimization problem \eqref{eq:mle}, with $\overline{\ten{\eta}} = 0$ we initialize the parameters using a simple heuristic approach. First, we compute moment based mode-wise marginal covariance estimates $\widehat{\mat{\Sigma}}_k(\ten{X})$ and $\widehat{\mat{\Sigma}}_k(\ten{F}_Y)$ as
\begin{displaymath}
    \widehat{\mat{\Sigma}}_k(\ten{X})   = \frac{1}{n}\sum_{i = 1}^{n} (\ten{X}_i)_{(k)}\t{(\ten{X}_i)_{(k)}}, \quad
    \widehat{\mat{\Sigma}}_k(\ten{F}_Y) = \frac{1}{n}\sum_{i = 1}^{n} (\ten{F}_{y_i})_{(k)}\t{(\ten{F}_{y_i})_{(k)}}.
\end{displaymath}
Then, for every mode $k = 1, \ldots, r$, we compute the first $j = 1, \ldots, q_k$ eigenvectors $\mat{v}_j(\widehat{\mat{\Sigma}}_k(\ten{X}))$, $\mat{v}_j(\widehat{\mat{\Sigma}}_k(\ten{F}_Y))$ and eigenvalues $\lambda_j(\widehat{\mat{\Sigma}}_k(\ten{X}))$, $\lambda_j(\widehat{\mat{\Sigma}}_k(\ten{X}))$ of the marginal covariance estimates. We set 
\begin{align*}
    \mat{U}_k &= (\mat{v}_1(\widehat{\mat{\Sigma}}_1(\ten{X})), \ldots, \mat{v}_{q_k}(\widehat{\mat{\Sigma}}_{q_k}(\ten{X}))), \\
    \mat{D}_k &= \diag(\mat{v}_1(\widehat{\mat{\Sigma}}_1(\ten{X}))\mat{v}_1(\widehat{\mat{\Sigma}}_1(\ten{F}_{Y})), \ldots, \mat{v}_{q_k}(\widehat{\mat{\Sigma}}_{q_k}(\ten{X}))\mat{v}_{q_k}(\widehat{\mat{\Sigma}}_k(\ten{F}_{Y}))), \\
    \mat{V}_k &= (\mat{v}_1(\widehat{\mat{\Sigma}}_1(\ten{F}_Y), \ldots, \mat{v}_{q_k}(\widehat{\mat{\Sigma}}_{q_k}(\ten{F}_Y)). \\
\end{align*}
The initial value of $\mat{\beta}_k$ is
\begin{displaymath}
    \hat{\mat{\beta}}_k^{(0)} = \mat{U}_k\sqrt{\mat{D}_k}\t{\mat{V}_k},
\end{displaymath}
and the initial value of $\mat{\Omega}_k$ is set to the identity $\mat{\Omega}_k^{(0)} = \mat{I}_{p_k}$, for $k=1,\ldots,r$.

Given $\hat{\mat{\beta}}_1, \ldots, \hat{\mat{\beta}}_r, \hat{\mat{\Omega}}_1, \ldots, \hat{\mat{\Omega}}_r$, we take the gradient $\nabla_{\mat{\beta}_j}l_n$ of the tensor normal log-likelihood $l_n$ in \eqref{eq:log-likelihood} applying \cref{thm:grad} and keep all other parameters except $\mat{\beta}_j$ fixed. Then, $\nabla_{\mat{\beta}_j}l_n = 0$ has the closed form solution
\begin{equation}\label{eq:tensor_normal_beta_solution}
    \t{\mat{\beta}_j} = \biggl(
        \sum_{i = 1}^{n}
        \Bigl( \ten{F}_{y_i}\mlm_{k \neq j}\hat{\mat{\Omega}}_k^{-1}\hat{\mat{\beta}}_k \Bigr)_{(j)}
        \t{\Bigl( \ten{F}_{y_i}\mlm_{k \neq j}\hat{\mat{\beta}}_k \Bigr)_{(j)}}
    \biggr)^{-1}
    \biggl(
        \sum_{i = 1}^{n}
        \Bigl( \ten{F}_{y_i}\mlm_{k \neq j}\hat{\mat{\beta}}_k \Bigr)_{(j)}
        \t{(\ten{X}_{i})_{(j)}}
    \biggr)
        \hat{\mat{\Omega}}_j.
\end{equation}
Equating the partial gradient of the $j$th scatter matrix $\mat{\Omega}_j$ in \cref{thm:grad} to zero ( $\nabla_{\mat{\Omega}_j}l_n = 0$) gives a quadratic matrix equation. This is due to the dependence of $\ten{\mu}_y$ on $\mat{\Omega}_j$. In practice though, it is faster, more stable, and equally accurate to use mode-wise covariance estimates via the residuals
\begin{displaymath}
    \hat{\ten{R}}_i = \ten{X}_i - \hat{\ten{\mu}}_{y_i} = \ten{X}_i - \ten{F}_{y_i}\mlm_{k = 1}^{r}\hat{\mat{\Omega}}_k^{-1}\hat{\mat{\beta}}_k.
\end{displaymath}
The estimates are computed via
\begin{displaymath}
    \tilde{\mat{\Sigma}}_j = \sum_{i = 1}^n (\hat{\ten{R}}_i)_{(j)} \t{(\hat{\ten{R}}_i)_{(j)}},
\end{displaymath}
where $\tilde{s}\tilde{\mat{\Sigma}}_j = \hat{\mat{\Omega}}_j^{-1}$. For scaling we use that the mean squared error has to be equal to the trace of the covariance estimate,
\begin{displaymath}
    \frac{1}{n}\sum_{i = 1}^n \langle \hat{\ten{R}}_i, \hat{\ten{R}}_i \rangle = \tr\bigkron_{k = r}^{1}\hat{\mat{\Omega}}_k^{-1} = \prod_{k = 1}^{r}\tr{\hat{\mat{\Omega}}_k^{-1}} = \tilde{s}^r\prod_{k = 1}^{r}\tr{\tilde{\mat{\Sigma}}_k},
\end{displaymath}
so that
\begin{displaymath}
    \tilde{s} = \biggl(\Bigl(\prod_{k = 1}^{r}\tr{\tilde{\mat{\Sigma}}_k}\Bigr)^{-1}\frac{1}{n}\sum_{i = 1}^n \langle \hat{\ten{R}}_i, \hat{\ten{R}}_i \rangle\biggr)^{1 / r}
\end{displaymath}
resulting in the estimates $\hat{\mat{\Omega}}_j = (\tilde{s}\tilde{\mat{\Sigma}}_j)^{-1}$.
Estimation is then performed by updating  the estimates $\hat{\mat{\beta}}_j$ via \eqref{eq:tensor_normal_beta_solution} for $j = 1, \ldots, r$, and then recompute the $\hat{\mat{\Omega}}_j$ estimates simultaneously keeping the $\hat{\mat{\beta}}_j$'s fixed. This procedure is repeated until convergence. % Convergence is very fast, experiments showed that convergence occures usualy in less than $10$ iterations.

A technical detail for numerical stability is to ensure that the scaled values $\tilde{s}\tilde{\mat{\Sigma}}_j$, assumed to be symmetric and positive definite, are well conditioned. Thus, we estimate the condition number of $\tilde{s}\tilde{\mat{\Sigma}}_j$ prior to computing the inverse. In case of ill-conditioning, we use the regularized $\hat{\mat{\Omega}}_j = (\tilde{s}\tilde{\mat{\Sigma}}_j + 0.2 \lambda_{1}(\tilde{s}\tilde{\mat{\Sigma}}_j)\mat{I}_{p_j})^{-1}$ instead, where $\lambda_{1}(\tilde{s}\tilde{\mat{\Sigma}}_j)$ is the first (maximum) eigenvalue. Experiments showed that this regularization is usually only required in the first few iterations.

Furthermore, if the parameter space follows a more general setting as in \cref{thm:param-manifold}, updating may produces estimates outside the parameter space. A simple and efficient method is to project every updated estimate onto the corresponding manifold.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Ising Model}\label{sec:ising_estimation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The general distribution of a binary vector is modeled by the \emph{multi-variate Bernoulli distribution} (see: \textcite{GraphicalModels-Whittaker2009, MVB-Dai2012, MVB-DaiDingWahba2013}). The \emph{Ising model} \textcite{Ising-Ising1924} is a special case, considering only two-way interactions. Its probability mass function (PMF) for a binary random vector $X\in\{ 0, 1 \}^p$ with natural parameters $\mat{\gamma}\in\mathbb{R}^{p(p + 1) / 2}$ is given by
\begin{displaymath}
    P_{\mat{\gamma}}(\mat{x}) = p_0(\mat{\gamma})\exp(\t{\vech(\mat{x}\t{\mat{x}})}\mat{\gamma}).
\end{displaymath}
The scaling factor $p_0(\mat{\gamma})\in\mathbb{R}_{+}$ ensures that $P_{\mat{\gamma}}$ is a PMF. It is equal to the probability of the zero event $P(X = \mat{0}) = p_0(\mat{\gamma})$. More commonly known as the \emph{partition function}, the reciprocal of $p_0$, given by
\begin{equation}\label{eq:ising-partition-function}
    p_0(\mat{\gamma})^{-1} = \sum_{\mat{x}\in\{0, 1\}^p}\exp(\t{\vech(\mat{x}\t{\mat{x}})}\mat{\gamma}).
\end{equation}
By an abuse of notation, let $\mat{\gamma}_{j l}$ address the element of $\mat{\gamma}$ corresponding to $\mat{x}_j\mat{x}_l$ in $\vech(\mat{x}\t{\mat{x}})$\footnote{Specifically, the element $\mat{\gamma}_{j l}$ of $\mat{\gamma}$ is a short hand for $\mat{\gamma}_{\iota(j, l)}$ with $\iota(j, l) = (\min(j, l) - 1)(2 p - \min(j, l)) / 2 + \max(j, l)$ mapping the matrix row index $j$ and column index $l$ to the corresponding half vectorization indices $\iota(j, l)$.}. The ``diagonal'' parameter $\mat{\gamma}_{j j}$ expresses the conditional log odds of $X_j = 1\mid X_{-j} = \mat{0}$, where the negative subscript in $X_{-j}$ describes the $p - 1$ dimensional vector $X$ with the $j$'t element removed. The ``off diagonal'' parameters $\mat{\gamma}_{j l}$, for $j\neq l$, are equal to the conditional log odds of simultanious occurence $X_j = 1, X_l = 1 \mid X_{-j, -l} = \mat{0}$. More precise, for $j\neq l$, the conditional probabitities and the natural parameters are related by
\begin{align}
    \mat{\gamma}_{j j} &= \log\frac{P_{\mat{\gamma}}(X_j = 1\mid X_{-j} = \mat{0})}{1 - P_{\mat{\gamma}}(X_j = 1\mid X_{-j} = \mat{0})}, \nonumber \\
    \mat{\gamma}_{j l} &= \log\frac{1 - P_{\mat{\gamma}}(X_j = 1\mid X_{-j} = \mat{0})P_{\mat{\gamma}}(X_l = 1\mid X_{-l} = \mat{0})}{P_{\mat{\gamma}}(X_j = 1\mid X_{-j} = \mat{0})P_{\mat{\gamma}}(X_l = 1\mid X_{-l} = \mat{0})}\frac{P_{\mat{\gamma}}(X_j = 1, X_l = 1\mid X_{-j, -l} = \mat{0})}{1 - P_{\mat{\gamma}}(X_j = 1, X_l = 1\mid X_{-j, -l} = \mat{0})} \label{eq:ising-two-way-log-odds}.
\end{align}

Conditional Ising models, incorporating the information of covariates $Y$ into the model, have also been considered \textcite{sparseIsing-ChengEtAt2014, sdr-mixedPredictors-BuraForzaniEtAl2022}. The direct way is to parameterize the parameter $\mat{\gamma} = \mat{\gamma}_y$ by the covariate $Y = y$ to model a conditional distribution $P_{\mat{\gamma}_y}(\mat{x}\mid Y = y)$.

We extend the conditional PMF by allowing the binary variables to be tensor values, that is for $\ten{X}\in\{ 0, 1 \}^{p_1\times\cdots\times p_r}$ we set $\mat{x} = \vec{\ten{X}}$, with dimension $p = \prod_{k = 1}^{r}p_k$. Considering the tensor structure of $\ten{X}$ is done by assuming Kronecker product constraints to the parameter vector $\mat{\gamma}_y$ in a similar fashion as for the tensor normal model. This means that we compair the PMF $P_{\mat{\gamma}_y}(\vec{\ten{X}} | Y = y)$ to the quadratic exponential family \eqref{eq:quadratic-exp-fam} with the natural parameters modeled by \eqref{eq:eta1} and \eqref{eq:eta2}. A detail to be considered is that the diagonal of $(\vec{\ten{X}})\t{(\vec{\ten{X}})}$ is equal to $\vec{\ten{X}}$. This gives the GMLM model as
\begin{align}
    P_{\mat{\gamma}_y}(\ten{X} \mid Y = y)
        &= p_0(\mat{\gamma}_y)\exp(\t{\vech((\vec{\ten{X}})\t{(\vec{\ten{X}})})}\mat{\gamma}_y) \nonumber \\
        &= p_0(\mat{\gamma}_y)\exp\Bigl(\Bigl\langle \ten{X}, \ten{F}_y\mlm_{k = 1}^{r}\mat{\beta}_k \Bigr\rangle + \Bigl\langle\ten{X}\mlm_{k = 1}^{r}\mat{\Omega}_k, \ten{X}\Bigr\rangle\Bigr)\label{eq:ising-cond-prob}
\end{align}
where we set $\overline{\ten{\eta}} = 0$ and $\mat{T}_2$ to the identity. This is an additional constraint to the model, the reason for $\overline{\ten{\eta}} = 0$ is that the diagonal elements of $\mat{\Omega} = \bigkron_{k = r}^{1}\mat{\Omega}_k$ take the role of $\overline{\ten{\eta}}$, althoug not fully. Having the diagonal of $\mat{\Omega}$ and $\overline{\ten{\eta}}$ handling the self interaction effects might lead to interference in the optimization routine. Another approach would be to use the $\mat{T}_2$ matrix to set the corresponding diagonal elements of $\mat{\Omega}$ to zero and let $\overline{\ten{\eta}}$ handle the self interaction effect. In \cref{sec:chess} we actually use $\mat{T}_2$ to enforce zero constants for specific elments of $\mat{\Omega}$. But for simplicity, we set $\mat{T}_2$ to the identity in the following. All of those approaches, namely setting $\overline{\ten{\eta}} = 0$, keeping $\overline{\ten{\eta}}$ or using $\mat{T}_2$, are theoretically solid and compatible with \cref{thm:grad,thm:param-manifold,thm:asymptotic-normality-gmlm}, assuming all axis dimensions $p_k$ are non-degenerate, that is $p_k > 1$ for all $k = 1, \ldots, r$. Regardles, under our modeling choise we get the relation between the natural parameters $\mat{\gamma}_y$ of the conditional Ising model and the GMLM parameters $\mat{\beta}_k$ and $\mat{\Omega}_k$ as
\begin{equation}\label{eq:ising-natural-params}
    % \t{\pinv{\mat{D}_p}}\mat{\gamma}_y
    %     = \vec(\mat{\Omega} + \diag(\mat{B}\vec{\ten{F}_y}))
    %     = \vec\Biggl(\bigkron_{k = r}^{1}\mat{\Omega}_k + \diag\biggl(\vec\Bigl(\ten{F}_y\mlm_{k = 1}^{r}\mat{\beta}_k\Bigr)\biggr)\Biggr).
    \mat{\gamma}_y
        = \t{\mat{D}_p}\vec(\mat{\Omega} + \diag(\mat{B}\vec{\ten{F}_y}))
        = \t{\mat{D}_p}\vec\Biggl(\bigkron_{k = r}^{1}\mat{\Omega}_k + \diag\biggl(\vec\Bigl(\ten{F}_y\mlm_{k = 1}^{r}\mat{\beta}_k\Bigr)\biggr)\Biggr).
\end{equation}

In contract to the tensor normal GMLM, the matrices $\mat{\Omega}_k$ are only required to be symmetric. More specificaly, we require $\mat{\Omega}_k$, for $k = 1, \ldots, r$, to be elements of an embedded submanifold of $\SymMat{p_k}$ (see: \cref{sec:kron-manifolds,sec:matrix-manifolds}). The mode wise reduction matrices $\mat{\beta}_k$ need to be elements of an embedded submanifold of $\mathbb{R}^{p_k\times q_k}$. Common choises are listed in \cref{sec:matrix-manifolds}. \todo{check if we need to exclude zero here!}

To solve the optimization problem \eqref{eq:mle}, given a data set $(\ten{X}_i, y_i)$, for $i = 1, \ldots, n$, we use a variation of gradient descent.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Initial Values}

The first step is to get reasonable starting values. Experiments showed that a good starting value of $\mat{\beta}_k$, for $k = 1, \ldots, r$, it to use the tensor normal estimates from \cref{sec:tensor_normal_estimation}, interprating $\ten{X}_i$ as continuous. For initial values of $\mat{\Omega}_k$, a different approach is required. Setting everything to the uninformed initial value, that is $\mat{\Omega}_k = \mat{0}$ as this corresponds to the conditional log odds to be $1:1$ for every component and pairwaide interaction. This is not possible, since $\mat{0}$ is a stationary point of the log-likelihood. This is directly observed by taking a look at the partial gradients of the log-likelihood in \cref{thm:grad}. Instead, we use a crude heuristic which threads every mode seperately and ignores any relation to the covariates. It is computationaly cheap and better than any of the alternatives we considered. For every $k = 1, \ldots, r$, let the $k$'th mode second moment estimate be
\begin{equation}\label{eq:ising-mode-moments}
    \hat{\mat{M}}_{2(k)} = \frac{p_k}{n p}\sum_{i = 1}^n (\ten{X}_i)_{(k)}\t{(\ten{X}_i)_{(k)}}
\end{equation}
which contains the $k$'th mode first moment estimate in its diagonal $\hat{\mat{M}}_{1(k)} = \diag\hat{\mat{M}}_{2(k)}$. Considering every column of the matricized observation $(\ten{X}_i)_{(k)}$ as a $p_k$ dimensional observation itself. The number of those artifically generated observations is $n \prod_{j\neq k}p_j$. Let $Z_k$ denote the random variable those artifical observations are realization of. Then, we can interprate the elements $(\hat{\mat{M}}_{1(k)})_{j}$ as the estimates of the probability $P((Z_k)_j = 1)$, that is the marginal probability of the $j$th element of $Z_k$ being $1$. Similar, for $l \neq j$ we have $(\hat{\mat{M}}_{2(k)})_{j l}$ estimating $P((Z_k)_j = 1, (Z_k)_l = 1)$, the marginal probability of two-way interactions. % Without any regard of accuracy ...
Now, we set the diagonal elements of $\mat{\Omega}_k$ to zero. For the off diagonal elements of $\mat{\Omega}_k$, we equate the conditional probabilities $P((Z_k)_j = 1 \mid (Z_k)_{-j} = \mat{0})$ and $P((Z_k)_j = 1, (Z_k)_l = 1\mid (Z_k)_{-j, -l} = \mat{0})$ with the marginal probability estimates $(\hat{\mat{M}}_{1(k)})_{j}$ and $(\hat{\mat{M}}_{2(k)})_{j l}$, respectively. Use \eqref{eq:ising-two-way-log-odds} then gives the initial estimates $\hat{\mat{\Omega}}_k^{(0)}$, with $j \neq l$ component wise
\begin{equation}\label{eq:ising-init-Omegas}
    (\hat{\mat{\Omega}}_k^{(0)})_{j j} = 0,
    \qquad
    (\hat{\mat{\Omega}}_k^{(0)})_{j l} = \log\frac{1 - (\hat{\mat{M}}_{1(k)})_{j}(\hat{\mat{M}}_{1(k)})_{l}}{(\hat{\mat{M}}_{1(k)})_{j}(\hat{\mat{M}}_{1(k)})_{l}}\frac{(\hat{\mat{M}}_{2(k)})_{j l}}{1 - (\hat{\mat{M}}_{2(k)})_{j l}}.
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Gradient Optimization}
Given initial values, the gradients provided by \cref{thm:grad} can be evaluated for the Ising model. The first step therefore is to determin the values of the inverse link components $\ten{g}_1(\mat{\gamma}_y) = \E[\ten{X} \mid Y = y]$ and $\ten{G}_2(\mat{\gamma}_y) = \ten{g}_2(\mat{\gamma}_y) = \E[\ten{X}\circ\ten{X} \mid Y = y]$. An imediate simplification is that the first moment is a part of the second moment. Its values are determined via $\vec(\E[\ten{X} \mid Y = y]) = \diag(\E[\ten{X}\circ\ten{X} \mid Y = y]_{(1, \ldots, r)})$. This means only the second moment needs to be computed, or estimated (see: \cref{sec:ising-bigger-dim}) in the case of slightly bigger $p$. For the Ising model, the conditional second moment with parameters $\mat{\gamma}_y$ is given by the matricized relation
\begin{equation}\label{eq:ising-m2}
    \ten{g}_2(\ten{\gamma}_y)_{(1, \ldots, r)} = \E[(\vec{\ten{X}})\t{(\vec{\ten{X}})}\mid Y = y] = p_0(\mat{\gamma}_y)\sum_{\mat{x}\in\{0, 1\}^{p}}\mat{x}\t{\mat{x}}\exp(\t{\vech(\mat{x}\t{\mat{x}})}\mat{\gamma}_y).
\end{equation}
The natural parameter $\mat{\gamma}_y$ is evaluated via \eqref{eq:ising-natural-params} enabeling us to compute the partial gradients of the log-likelihood $l_n$ \eqref{eq:log-likelihood} for the Ising model by \cref{thm:grad} for the GMLM parameters $\mat{\beta}_k$ and $\mat{\Omega}_k$, $k = 1, \ldots, r$, at the current iterate $\mat{\theta}^{(I)} = (\mat{\beta}_1^{(I)}, \ldots, \mat{\beta}_r^{(I)}, \mat{\Omega}_1^{(I)}, \ldots, \mat{\Omega}_r^{(I)})$. Using classic gradient ascent for maximizing the log-likelihood, we have to specify a learning rate $\lambda\in\mathbb{R}_{+}$, usualy something like $10^{-3}$. The update rule is
\begin{displaymath}
    \mat{\theta}^{(I + 1)} = \mat{\theta}^{(I)} + \lambda\nabla_{\mat{\theta}} l_n(\mat{\theta})\bigr|_{\mat{\theta} = \mat{\theta}^{(I)}},
\end{displaymath}
which is iterated till convergence. In practice, iteration is performed until ether a maximum number of iterations is exhausted and/or some break condition is satisfied. A proper choise of the learning rate is needed as a too large learning rate $\lambda$ causes instabilities, while a too low learning rate requires an enourmes ammount of iterations. Generically, there are two approach against the need to determine a proper lerning rate. First, \emph{line search methods} determin an appropriate step size for every iteration. This works great if the evaluation of the object function (the log-likelihood) is cheap. This is not the case in our setting, see \cref{sec:ising-bigger-dim}. The second approach is an \emph{addaptive learning rate}. The basic idea is to track specific statistics while optimizing and dynamiclly addapt the leaning rate via well tested heuristics using the gathered knowledge from past iterations. We opted to use an addaptive leaning rate approach, this not only levaites the need to determin an approriate leaning rate, but also excelerates learning.

Our method of choise is RMSprop, which stands for \emph{root mean squared propagation} \textcite{rmsprop-Hinton2012}. This is a well known method in maschine learning for training neural networks. Its a variation of gradient descent with an per scalar parameter addaptive learning rate. It tracks a moving average of the element wise squared gradient $\mat{g}_2^{(I)}$, which is then used to scale (element wise) the gradient in the update rule. See \textcite{rmsprop-Hinton2012,deeplearningbook-GoodfellowEtAl2016} among others. The update rule using RMSprop for maximization\footnote{Instead of the more common minimization, therefore $+$ in the update of $\mat{\theta}$.} is
\begin{align*}
    \mat{g}_2^{(I + 1)} &= \nu \mat{g}_2^{(I)} + (1 - \nu)\nabla l_n(\mat{\theta}^{(I)})\odot\nabla l_n(\mat{\theta}^{(I)}), \\
    \mat{\theta}^{(I + 1)} &= \mat{\theta}^{(I)} + \frac{\lambda}{\sqrt{\mat{g}_2^{(I + 1)}} + \epsilon}\odot\nabla l_n(\mat{\theta}^{(I)}).
\end{align*}
The parameters $\nu = 0.9$, $\lambda = 10^{-3}$ and $\epsilon\approx 1.49\cdot 10^{-8}$ are fixed. The initial value of $\mat{g}_2^{(0)} = \mat{0}$, the symbol $\odot$ denotes the Hadamard product, that is the element wise multiplication. The divition and sqaure root operation are performed element wise as well. According to our experiments, RMSprop requires in the range of $50$ till $1000$ iterations till convergence while gradient ascent with a learning rate of $10^{-3}$ is in the range of $1000$ till $10000$. \todo{check this!}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Small Data Sets}\label{sec:ising-small-data-sets}
In case of a finite number of observations, specifically in data sets with a small number of observations $n$, the situation where one components is always ether zero or one can occure. Its also possible to observe two exclusive components. This situation of a ``degenerate'' data set needs to be saveguarded against in practive. Working with parameters on a log-scale, this gives estimates of $\pm\infty$. This is outside of the parameter space and breaks our optimization algorithm.

The first situation where this needs to be addressed is in \eqref{eq:ising-init-Omegas}, where we set initial estimates for $\mat{\Omega}_k$. To avoid divition by zero as well as evaluating the log of zero, we addapt \eqref{eq:ising-mode-moments}, the mode wise moment estimates $\hat{\mat{M}}_{2(k)}$. A simple method is to replace the ``degenerate'' components, that are entries with value $0$ or $1$, with the smallest positive estimate of exactly one occurence $p_k / n p$, or all but one occurence $1 - p_k / n p$, respectively.

The same problem arives in gradient optimization. Therefore, before starting the optimization, we detect degenerate combinations. We compute upper and lower bounds for the ``degenerate'' element in the Kronecker product $\hat{\mat{\Omega}} = \bigkron_{k = r}^{1}\hat{\mat{\Omega}}_k$. After every gradient update, we check if any of the ``degenerate'' elements fall outside of the bounds. In that case, we adjust all the elements of the Kronecker component estimates $\hat{\mat{\Omega}}_k$, corresponding to the ``degenerate'' element of their Kronecker product, to fall inside the precomputed bounds. While doing so, we try to alter every component as little as possible to ensure that the non-degenerate elements in $\hat{\mat{\Omega}}$, effected by this change due to its Kronecker structure, are altered as little as possible. The exact details are technically cumbersome while providing little insight. \todo{For more details we refer the reader to the source code prodived with the supplemental material.}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Slightly Bigger Dimensions}\label{sec:ising-bigger-dim}
A big challange with the Ising model is its high computational complexity. The reason is in the sum over all binary vectors of length $p = \prod_{k = 1}^{r}p_k$ in the partition function \eqref{eq:ising-partition-function}. Computing the partition function exactly requires to sum all $2^p$ binary vectors. Small dimensions, like $p\approx 10$, this is easily computed. Increasing the dimension bejond $20$ gets extremely expensive while for dimensions bigger than $30$ its absolutely impossible. Trying to avoid the evaluation of the log-likelihood and only computing its partial gradients via \cref{thm:grad} does not resolve the issue. The gradients require the inverse link, in other words the second moment \eqref{eq:ising-m2}, where, if dropping the scaling factor $p_0$, still involves to sum $2^p$ summands. Basically, with our model, this means that the optimization of the Ising model using exactly computed gradients is impossible for moderately sized problems.

For estimation of dimensions $p$ bigger than $20$, we use a Monte-Carlo method to estimate the second moment \eqref{eq:ising-m2}, required to compute the partial gradients of the log-likelihood. Specifically, we use a Gibbs-Sampler to sample from the conditional distribution and approximate the second moment in an importance sampling framework. This can be implemented quite efficiently while the estimation accuracy for the second moment is evaluated experimentaly which seems to be very reliable. Simultaniously, we use the same approach to estimate the partition function. %This though, is in comparison inaccurate, and may only be used to get a rough idea of the log-likelihood. Regardles, for our method, we only need the gradient for optimization where appropriate break conditions, not based on the likelihood, lead to a working method for MLE estimation.

\begin{figure}
    \centering
    \includegraphics[]{plots/sim-ising-perft-m2.pdf}
    \caption{\label{fig:ising-m2-perft}Performance test for computing/estimating the second moment of the Ising model of dimension $p$ using ether the exact method or a Monte-Carlo simulation.}
\end{figure}


\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Statistical Properties}

\subsection{Kronecker Product Manifolds}\label{sec:kron-manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\cref{thm:sdr} identifies the sufficient reduction for the regression of $Y$ on $\ten{X}$ in the population. Any estimation of the sufficient reduction requires application of some optimality criterion. As we operate within the framework of the exponential family, we opted for maximum likelihood estimation (MLE). For the unconstrained problem, where the parameters are simply $\mat{B}$ and $\mat{\Omega}$ in \eqref{eq:eta1-manifold}, maximizing the likelihood of $\ten{X} \mid Y$ is straightforward and yields well-defined MLEs of both parameters. Our setting, though, requires the constrained optimization of the $\ten{X} \mid Y$ likelihood subject to $\mat{B} = \bigotimes_{j = r}^{1}\mat{\beta}_j$ and $\mat{\Omega}=\bigkron_{j = r}^{1}\mat{\Omega}_j$. \Cref{thm:kron-manifolds,thm:param-manifold} provide the setting for which the MLE of the constrained parameter $\mat{\theta}$ is well-defined, which in turn leads to the derivation of its asymptotic normality.

The main problem to be solves for asymtotic results of the MLE of the constraint parameter $\mat{\theta} = (\overline{\ten{\eta}}, \vec\mat{B}, \vech\mat{\Omega})$ derives from the nature of the constraint. We assumed that $\mat{B} = \bigkron_{k = r}^{1}\mat{\beta}_k$, where the parameter $\mat{B}$ is identifiable. This means that different values of $\mat{B}$ lead to different densities $f_{\mat{\theta}}(\ten{X}\mid Y = y)$, a basic property needed to ensure consistency of parameter estimates, which in turn is needed for asymptotic normality. On the other hand, the components $\mat{\beta}_j$ for $j = 1, \ldots, r$ are \emph{not} identifyable. A direct consequence of the identity $\mat{\beta}_2\otimes\mat{\beta}_1 = (c\mat{\beta}_2)\otimes (c^{-1}\mat{\beta}_1)$ for every $c\neq 0$. This is the reason we formulated $\Theta$ as a constraint parameter space instead of parameterizing the densities of $\ten{X}\mid Y$ with respect to the components $\mat{\beta}_1, \ldots, \mat{\beta}_r$. The situation for $\mat{\Omega} = \bigkron_{k = r}^{1}\mat{\Omega}_k$ is the same.

In addition to identifyable parameters, asymtotic normality given in \cref{thm:asymptotic-normality-gmlm} requires the notion of differentiation. Therefore, the space itself needs to be able to define differentiation, usually a vector space. This is a too strong assumption for our purposes. To weaken the vector space assumption we considure \emph{smooth manifolds}. Basically, spaces which localy look like Euclidean space and allow the notion of differentiation. The more general \emph{topological} manifolds are too weak for differentiation. To make matters worse, a smooth manifold only allows first derivatives. Without goint into any details, a \emph{Riemannian manifold} is the solution. Similar to an abstract \emph{smooth manifold}, those spaces are detached from our usual intuition as well as complicated to handle in an already complicated setting. This is where an \emph{embedded (sub)manifold} comes to rescue. Simply speaking, an embedded manifold is a manifold which is a subset of a manifold from which it inherits its properties. Now, if a manifold is embedded in Euclidean space, almost all the complication of the abstract manifold theory simplifies drastically. Moreover, since Euclidean space is itself a Riemannian manifold, we inherit the means for higher derivatives. Finally, smooth embedded submanifold structure for the parameter space maintains consistency with existing approaches and results for parameter sets with linear subspace structure. Those reasons are the argument to ensure that the constraint parameter space $\Theta$ is an \emph{smooth embedded submanifold} in an open subset $\Xi$ of Euclidean space.

Now, we diretly define a \emph{smooth manifold} embedded in $\mathbb{R}^p$ without any detours to the more generel theory. See for example \textcite{introToSmoothMani-Lee2012,,introToRiemannianMani-Lee2018,optimMatrixMani-AbsilEtAl2007,aufbauAnalysis-kaltenbaeck2021} among others.
\begin{definition}[Manifolds]\label{def:manifold}
    A set $\manifold{A}\subseteq\mathbb{R}^p$ is an \emph{embedded smooth manifold} of dimension $d$ if for every $\mat{x}\in\manifold{A}$ there exists a smooth\footnote{Here \emph{smooth} means infinitely differentiable or $C^{\infty}$.} bi-continuous map $\varphi:U\cap\manifold{A}\to V$, called a \emph{chart}, with $\mat{x}\in U\subseteq\mathbb{R}^p$ open and $V\subseteq\mathbb{R}^d$ open.
\end{definition}

We also need the concept of a \emph{tangent space} to formulate asymtotic normality in a way which is independent of a particular coordinate representation. Intuitively, the tangent space at a point $\mat{x}\in\manifold{A}$ of the manifold $\manifold{A}$ is the hyperspace of all velocity vectors $\t{\nabla\gamma(0)}$ of any curve $\gamma:(-1, 1)\to\manifold{A}$ passing through $\mat{x} = \gamma(0)$, see \cref{fig:torus}. Locally, at $\mat{x} = \gamma(0)$ with a chart $\varphi$ we can written $\gamma(t) = \varphi^{-1}(\varphi(\gamma(t)))$ which gives that $\Span\t{\nabla\gamma(0)} \subseteq \Span\t{\nabla\varphi^{-1}(\varphi(\mat{x}))}$. Taking the union over all smooth curves through $\mat{x}$ gives equality. The following definition leverages the simplified setup of smooth manifolds in Euclidean space.

\begin{definition}[Tangent Space]\label{def:tangent-space}
    Let $\manifold{A}\subseteq\mathbb{R}^p$ be an embedded smooth manifold and $\mat{x}\in\manifold{A}$. The \emph{tangent space} at $\mat{x}$ of $\manifold{A}$ is defined as
    \begin{displaymath}
        T_{\mat{x}}\manifold{A} := \Span\t{\nabla\varphi^{-1}(\varphi(\mat{x}))}
    \end{displaymath}
    for any chart $\varphi$ with $\mat{x}$ in the pre-image of $\varphi$.
\end{definition}

\Cref{def:tangent-space} is consistent since it can be shown that two different charts at the same point have identical span.

\begin{figure}
    \centering
    \includegraphics[width = 0.5\textwidth]{images/TorustangentSpace.pdf}
    \caption{\label{fig:torus}Visualization of the tangent space $T_{\mat{x}}\manifold{A}$ at $\mat{x}$ of the torus $\manifold{A}$. The torus $\manifold{A}$ is a 2-dimensional embedded manifold in $\mathbb{R}^3$. The tangent space $T_{\mat{x}}\manifold{A}\subset\mathbb{R}^3$ is a the 2-dimensional hyperplane visualized with its origin $\mat{0}$ shifted to $\mat{x}$. Moreover, two curves $\gamma_1, \gamma_2$ on the torus are drawn with $\mat{x} = \gamma_1(0) = \gamma_2(0)$. The curve velocity vectors $\t{\nabla\gamma_1(0)}$ and $\t{\nabla\gamma_2(0)}$ are drawn as tangent vectors with root $\mat{x}$.}
\end{figure}




As a basis to ensure that the constraint parameter space $\Theta$ is a manifold, which is the statement of \cref{thm:param-manifold}, we need \cref{thm:kron-manifolds}. Therefore, we need the notion of a \emph{spherical} set, which is a set $\manifold{A}$, on which the Frobenius norm is constant. That is, $\|\,.\,\|_F:\manifold{A}\to\mathbb{R}$ is constant. Forthermore, we call a scale invariant set $\manifold{A}$ a \emph{cone}, that is $\manifold{A} = \{ c \mat{A} : \mat{A}\in\manifold{A} \}$ for all $c > 0$.

\begin{theorem}[Kronecker Product Manifolds]\label{thm:kron-manifolds}
    Let $\manifold{A}\subseteq\mathbb{R}^{p_1\times q_1}\backslash\{\mat{0}\}, \manifold{B}\subseteq\mathbb{R}^{p_2\times q_2}\backslash\{\mat{0}\}$ be smooth embedded submanifolds. Assume one of the following conditions holds.
    \begin{itemize}
        \item[-] ``sphere condition'':
            At least one of $\manifold{A}$ or $\manifold{B}$ is \emph{spherical} and let $d = \dim\manifold{A} + \dim\manifold{B}$.
        \item[-] ``cone condition'':
            Both $\manifold{A}$ and $\manifold{B}$ are \emph{cones} and let $d = \dim\manifold{A} + \dim\manifold{B} - 1$.
    \end{itemize}
    Then, $\{ \mat{A}\otimes \mat{B} : \mat{A}\in\manifold{A}, \mat{B}\in\manifold{B} \}\subset\mathbb{R}^{p_1 p_2\times q_1 q_2}$ is a smooth embedded $d$-manifold.
\end{theorem}

\begin{theorem}[Parameter Manifold]\label{thm:param-manifold}
    Let
    \begin{displaymath}
        \manifold{K}_{\mat{B}} = \Bigl\{ \bigkron_{k = r}^{1}\mat{\beta}_k : \mat{\beta}_k\in\manifold{B}_k \Bigr\}
        \quad\text{and}\quad
        \manifold{K}_{\mat{\Omega}} = \Bigl\{ \bigkron_{k = r}^{1}\mat{\Omega}_k : \mat{\Omega}_k\in\manifold{O}_k \Bigr\}
    \end{displaymath}
    where $\manifold{B}_k\subset\mathbb{R}^{p_k\times q_k}\backslash\{\mat{0}\}$ and $\manifold{O}_k\subset\mathbb{R}^{p_k\times p_k}\backslash\{\mat{0}\}$ are smooth embedded manifolds which are ether spheres or cones, for $k = 1, ..., r$. Furthermore, let
    \begin{displaymath}
        \manifold{CK}_{\mat{\Omega}} = \{ \vech{\mat{\Omega}} : \mat{\Omega}\in\manifold{K}_{\mat{\Omega}} \land \pinv{(\mat{T}_2\pinv{\mat{D}_p})}\mat{T}_2\pinv{\mat{D}_p}\vec{\mat{\Omega}} = \vec{\mat{\Omega}} \}
    \end{displaymath}
    then the constrained parameter space $\Theta = \mathbb{R}^p \times \manifold{K}_{\mat{B}}\times\manifold{CK}_{\mat{\Omega}}\subset\mathbb{R}^{p(p + 2 q + 3) / 2}$ is a smooth embedded manifold.
\end{theorem}

\subsection{Matrix Manifolds}\label{sec:matrix-manifolds}
A powerful side effect of \cref{thm:param-manifold} is the modeling flexibility it provides. For example, we can perform low rank regression. Or, we may constrain two-way interactions between direct axis neighbors by using band matrices for the $\mat{\Omega}_k$'s, among others.

This flexibility derives from many different matrix manifolds that can be used as building blocks $\manifold{B}_k$ and $\manifold{O}_k$ of the parameter space $\Theta$ in \cref{thm:param-manifold}. A list of possible choices, among others, is given in \cref{tab:matrix-manifolds}. As long as parameters in $\Theta$ are valid paramererization of a density (or PMF) of \eqref{eq:quadratic-exp-fam} subject to \eqref{eq:eta1-manifold} and \eqref{eq:eta2-manifold}, one may choose any of the manifolds listed in \cref{tab:matrix-manifolds} which are ether cones or spherical. We also included an example which is nether a sphere nor a cone. They may also be valid building blocks, but require more work as they are not directly leading to a parameter manifold by \cref{thm:param-manifold}. In case one can show the resulting parameter space $\Theta$ is an embedded manifold, the asymtotic theory of \cref{sec:asymtotics} is applicable.

\begin{table}
    \centering
    \begin{tabular}{l | l | c c c}
        Symbol & Description & C & S & Dimension\\ \hline
        $\mathbb{R}^{p\times q}$ & All matrices of dimension $p\times q$ &
            \checkmark & \xmark     & $p q$ \\ \hline
        $\mathbb{R}_{*}^{p\times q}$ & Full rank $p\times q$ matrices &
            \checkmark & \xmark     & $p q$                  \\ \hline
        $\Stiefel{p}{q}$ & \emph{Stiefel Manifold}, $\{ \mat{U}\in\mathbb{R}^{p\times q} : \t{\mat{U}}\mat{U} = \mat{I}_q \}$ for $q\leq p$ &
            \xmark     & \checkmark & $p q - q (q + 1) / 2$  \\ \hline
        $\mathcal{S}^{p - 1}$ & Unit sphere in $\mathbb{R}^p$, special case $\Stiefel{p}{1}$ &
            \xmark     & \checkmark & $p - 1$                \\ \hline
        $\UnitaryGrp{p}$ & Unitary Group, special case $\Stiefel{p}{p}$ &
            \xmark     & \checkmark & $p (p - 1) / 2$        \\ \hline
        $\SpecialUnitaryGrp{p}$ & Special Unitary Group $\{ \mat{U}\in U(p) : \det{\mat{U}} = 1 \}$ &
            \xmark     & \checkmark & $p (p - 1) / 2$        \\ \hline
        $\mathbb{R}_{r}^{p\times q}$ & Matrices of known rank $r > 0$, generalizes $\StiefelNonCompact{p}{q}$ &
            \checkmark & \xmark     & $r(p + q - r)$         \\ \hline
        & Symmetric matrice &
            \checkmark & \xmark     & $p (p + 1) / 2$    \\ \hline
        $\SymPosDefMat{p}$ & Symmetric Positive Definite matrices &
            \checkmark & \xmark     & $p (p + 1) / 2$    \\ \hline
        & Scaled Identity $\{ a\mat{I}_p : a\in\mathbb{R}_{+} \}$ &
            \checkmark & \xmark     & $1$    \\ \hline
        & Symmetric $r$-band matrices (includes diagonal) &
            \checkmark & \xmark     & $(2 p - r) (r + 1) / 2$    \\
            & $\qquad\{ \mat{A}\in\mathbb{R}^{p\times p} : \mat{A} = \t{\mat{A}}\land \mat{A}_{i j} = 0\ \forall |i - j| > r \}$ \\ \hline
        & Auto correlation alike $\{ \mat{A}\in\mathbb{R}^{p\times p} : \mat{A}_{i j} = \rho^{|i - j|}, \rho\in(0, 1) \}$ &
            \xmark     & \xmark     & $1$ \\ \hline
    \end{tabular}
    \caption{\label{tab:matrix-manifolds}Examples of embedded matrix manifolds. ``Symbol'' a (more or less) common notation for the matrix manifold, if at all. ``C'' stands for \emph{cone}, meaning it is scale invariant. ``S'' means \emph{spherical}, that is, constant Frobenius norm.}
\end{table}

\begin{remark}
    The \emph{Grassmann Manifold} of $q$ dimensional subspaces in $\mathbb{R}^p$ is not listed in \cref{tab:matrix-manifolds} since it is not embedded in $\mathbb{R}^{p \times q}$.
\end{remark}


\subsection{Asymptotics}\label{sec:asymtotics}

Let $Z$ be a random variable distributed according to a parameterized probability distribution with density $f_{\mat{\theta_0}}\in\{ f_{\mat{\theta}} : \mat{\theta}\in\Theta \}$ where $\Theta$ is a subset of Euclidean space. We want to estimate the parameter ${\mat{\theta}}_0$ using $n$ i.i.d. (independent and identically distributed) copies of $Z$. We assume a known, real-valued and measurable function $z\mapsto m_{\mat{\theta}}(z)$ for every $\mat{\theta}\in\Theta$ while ${\mat{\theta}}_0$ maximizes the map $\mat{\theta}\mapsto M(\mat{\theta}) = \E m_{\mat{\theta}}(Z)$ uniquely. For the estimation we maximize the empirical version
\begin{align}\label{eq:Mn}
    M_n(\mat{\theta}) &= \frac{1}{n}\sum_{i = 1}^n m_{\mat{\theta}}(Z_i).
\end{align}
An \emph{M-estimator} $\hat{\mat{\theta}}_n = \hat{\mat{\theta}}_n(Z_1, ..., Z_n)$ is a maximizer for the objective function $M_n$ over the parameter space $\Theta$ defined as
\begin{displaymath}
    \hat{\mat{\theta}}_n = \argmax_{\mat{\theta}\in\Theta} M_n(\mat{\theta}).
\end{displaymath}


It is not necessary to have a perfect maximizer, as long as the objective has finite supremum, it is sufficient to take an \emph{almost maximizer} $\hat{\mat{\theta}}_n$ as defined in the following;

\begin{definition}[weak and strong M-estimators]
    An estimator $\hat{\mat{\theta}}_n$ for the objective function $M_n$ in \eqref{eq:Mn} with $\sup_{\mat{\theta}\in\Theta}M_n(\mat{\theta}) < \infty$ such that
    \begin{displaymath}
        M_n(\hat{\mat{\theta}}_n) \geq \sup_{\mat{\theta}\in\Theta}M_n(\mat{\theta}) - o_P(n^{-1})
    \end{displaymath}
    is called a \emph{strong M-estimator} over $\Theta$. Replacing $o_P(n^{-1})$ by $o_P(1)$ gives a \emph{weak M-estimator}.
\end{definition}



\begin{theorem}[Asymptotic Normality]\label{thm:asymptotic-normality-gmlm}
    Assume $Z = (\ten{X}, Y)$ satisfies model \eqref{eq:quadratic-exp-fam} subject to \eqref{eq:eta1-manifold} and \eqref{eq:eta2-manifold} with true constrained parameter $\mat{\theta}_0 = (\overline{\eta}_0, \mat{B}_0, \mat{\Omega}_0)\in\Theta$, where $\Theta$ is defined in \cref{thm:param-manifold}. Under the regularity \crefrange{cond:differentiable-and-convex}{cond:finite-sup-on-compacta} in the appendix, there exists a strong M-estimator sequence $\hat{\mat{\theta}}_n$ deriving from $l_n$ in \eqref{eq:log-likelihood} over $\Theta$. Furthermore, any strong M-estimator $\hat{\mat{\theta}}_n$ converges in probability to the true parameter $\mat{\theta}_0$ over $\Theta$. That is, $ \hat{\mat{\theta}}_n\xrightarrow{p}\mat{\theta}_0$.  Moreover, every strong M-estimator $\hat{\mat{\theta}}_n$ is asymptotically normal,
    \begin{displaymath}
        \sqrt{n}(\hat{\mat{\theta}}_n - \mat{\theta}_0) \xrightarrow{d} \mathcal{N}_{p(p + 2 q + 3) / 2}(0, \mat{\Sigma}_{\mat{\theta}_0})
    \end{displaymath}
    with asymptotic variance-covariance structure $\mat{\Sigma}_{\mat{\theta}_0}$ given in \eqref{eq:asymptotic-covariance-gmlm}.
\end{theorem}


The following is a reformulation of \textcite[Lemma~2.3]{asymptoticMLE-BuraEtAl2018} which assumes Condition~2.2 to hold. But the existence of a mapping in Condition~2.2 is not needed for Lemma~2.3. It suffices that the restricted parameter space $\Theta$ is a subset of the unrestricted parameter space $\Xi$, which is trivially satisfied in our setting. Under this, \cref{thm:exists-strong-M-estimator-on-subsets} follows directly from \textcite[Lemma~2.3]{asymptoticMLE-BuraEtAl2018}.

\begin{theorem}[Existence of strong M-estimators on Subsets]\label{thm:exists-strong-M-estimator-on-subsets}
    Assume there exists a (weak/strong) M-estimator $\hat{\mat{\xi}}_n$ for $M_n$ over $\Xi$, then there exists a strong M-estimator $\hat{\mat{\theta}}_n$ for $M_n$ over any non-empty $\Theta\subseteq\Xi$.
\end{theorem}


\begin{theorem}[Existence and Consistency of M-estimators on Subsets]\label{thm:M-estimator-consistency-on-subsets}
    Let $\Xi$ be a convex open subset of a Euclidean space and $\Theta\subseteq\Xi$ non-empty. Assume $\mat{\xi}\mapsto m_{\mat{\xi}}(z)$ is a strictly concave function on $\Xi$ for almost all $z$ and $z\mapsto m_{\mat{\xi}}(z)$ is measurable for all $\mat{\xi}\in\Xi$. Furthermore, let $M(\mat{\xi}) = \E m_{\mat{\xi}}(Z)$ be a well defined function with a unique maximizer $\mat{\theta}_0\in\Theta\subseteq\Xi$, that is $M(\mat{\theta}_0) > M(\mat{\xi})$ for all $\mat{\xi}\neq\mat{\theta}_0$. Also, let for every non-empty compact $K\subset\Xi$
    \begin{displaymath}
        \E\sup_{\mat{\xi}\in K}|m_{\mat{\xi}}(Z)| < \infty.
    \end{displaymath}
    Then, there exists a strong M-estimator $\hat{\mat{\theta}}_n$ of $M_n(\mat{\theta}) = \frac{1}{n}\sum_{i = 1}^{n} m_{\mat{\theta}}(Z_i)$ over the subset $\Theta$. Moreover, any strong M-estimator $\hat{\mat{\theta}}_n$ of $M_n$ over $\Theta$ converges in probability to $\mat{\theta}_0$, that is $\hat{\mat{\theta}}_n\xrightarrow{p}\mat{\theta}_0$.
\end{theorem}



\todo{The assumptions of the following can be a bit weakened, is this neccessary? For example the Hessian can be singular but is non-singular constraint to the tangent space. We can also only define $\mat{\theta}\mapsto m_{\mat{\theta}}$ only on the manifold which makes the statement much more technical, but way more general while we need to ensure that every smooth local extention of $\mat{\theta}\mapsto m_{\mat{\theta}}$ yields the same statement, which it does, but well, then it gets more complicated! Maybe add these things as a remark? The even more general formulation for Riemannian Manifolds is definitely over the top!}

\begin{theorem}[Asymptotic Normality for M-estimators on Manifolds]\label{thm:M-estimator-asym-normal-on-manifolds}
    Let $\Theta\subseteq\mathbb{R}^p$ be a smooth embedded manifold. For each $\mat{\theta}$ in a neighborhood in $\mathbb{R}^p$ of the true parameter $\mat{\theta}_0\in\Theta$ let $z\mapsto m_{\mat{\theta}}(z)$ be measurable and $\mat{\theta}\mapsto m_{\mat{\theta}}(z)$ be differentiable at $\mat{\theta}_0$ for almost all $z$. Assume also that there exists a measurable function $u$ such that $\E[u(Z)^2] < \infty$, and for almost all $z$ as well as all $\mat{\theta}_1, \mat{\theta}_2$ in a neighborhood of $\mat{\theta}_0$ such that
    \begin{displaymath}
        | m_{\mat{\theta}_1}\!(z) - m_{\mat{\theta}_2}\!(z) | \leq u(z) \| \mat{\theta}_1 - \mat{\theta}_2 \|_2.
    \end{displaymath}
    Moreover, assume that $\mat{\theta}\mapsto\E[m_{\mat{\theta}}(Z)]$ admits a second-order Taylor expansion at $\mat{\theta}_0$ in a neighborhood of $\mat{\theta}_0$ in $\mathbb{R}^p$ with a non-singular Hessian $\mat{H}_{\mat{\theta}_0} = \nabla^2_{\mat{\theta}}\E[m_{\mat{\theta}}(Z)]|_{\mat{\theta} = \mat{\theta}_0}\in\mathbb{R}^{p\times p}$.

    If $\hat{\mat{\theta}}_n$ is a strong M-estimator of $\mat{\theta}_0$ in $\Theta$, then $\hat{\mat{\theta}}_n$ is asymptotically normal
    \begin{displaymath}
        \sqrt{n}(\hat{\mat{\theta}}_n - \mat{\theta}_0) \xrightarrow{d} \mathcal{N}_p(\mat{0}, \mat{\Pi}_{\mat{\theta}_0} \E[\nabla_{\mat{\theta}} m_{\mat{\theta}_0}(Z)\t{(\nabla_{\mat{\theta}} m_{\mat{\theta}_0}(Z))}]\mat{\Pi}_{\mat{\theta}_0})
    \end{displaymath}
    where $\mat{\Pi}_{\mat{\theta}_0} = \mat{P}_{\mat{\theta}_0}\pinv{(\t{\mat{P}_{\mat{\theta}_0}}\mat{H}_{\mat{\theta}_0}\mat{P}_{\mat{\theta}_0})}\t{\mat{P}_{\mat{\theta}_0}}$ and $\mat{P}_{\mat{\theta}_0}$ is any matrix whose span is the tangent space $T_{\mat{\theta}_0}\Theta$ of $\Theta$ at $\mat{\theta}_0$.
\end{theorem}


\begin{remark}
    \cref{thm:M-estimator-asym-normal-on-manifolds} has as special case Theorem~5.23 in \textcite{asymStats-van_der_Vaart1998}, when $\Theta$ is open subset of a Euclidean space as opposed to a smooth embedded manifold. 
    \todo{I don't like it that much, mention that an open set if an embedded manifold implying that $\mat{\Pi}_{\mat{\theta}_0} = \mat{H}_{\mat{\theta}_0}^{-1}$}
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simulations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we provide simulation results for the tensor normal as well as the Ising model where different aspects of the GMLM model are compaired against other methods. The comparison methods are TSIR \textcite{tsir-DingCook2015}, MGCCA \textcite{MGCCA-GirkaEtAl2024} and HOPCA \todo{cite?!} for both continuous and binary data. In case of binary data, the binary values are simply treated as continuous. As a base line we also include classic PCA on vectorized observations. \todo{check, fix, ...}

All experiments are performed with different sample sizes $n = 100, 200, 300, 500$ and $750$. Every experiment is repeated $100$ times.

We are interested in the quality of the estimate of the true sufficient reduction $\ten{R}(\ten{X})$ from \cref{thm:sdr}. Therefore, we compair against the true vectorized reduction matrix $\mat{B} = \bigkron_{k = r}^{1}\mat{\beta}_k$, since the vectorized version is compatible with any linear reduction method. The distance $d(\mat{B}, \hat{\mat{B}})$ between $\mat{B}\in\mathbb{R}^{p\times q}$ and an estimate $\hat{\mat{B}}\in\mathbb{R}^{p\times \tilde{q}}$ is the \emph{subspace distance} which is proportional to
\begin{displaymath}
    d(\mat{B}, \hat{\mat{B}}) \propto \| \mat{B}\pinv{(\t{\mat{B}}\mat{B})}\t{\mat{B}} - \hat{\mat{B}}\pinv{(\t{\hat{\mat{B}}}\hat{\mat{B}})}\t{\hat{\mat{B}}} \|_F.
\end{displaymath}
That is the Frobenius norm of the diffeence between the projections onto the span of $\mat{B}$ and $\hat{\mat{B}}$. The proportionality constant\footnote{Depends on row dimension $p$ and the ranks of $\mat{B}$ and $\hat{\mat{B}}$ given by $(\min(\rank\mat{B} + \rank\hat{\mat{B}}, 2 p - (\rank\mat{B} + \rank\hat{\mat{B}})))^{-1/2}$.} of $d(\mat{B}, \hat{\mat{B}})$ ensures that the subspace distance is in the interval $[0, 1]$. A distance of zero implies equality of the spans, a distance of one means that the subspaces are orthogonal.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Tensor Normal}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For every tensor normal model we draw i.i.d. samples $\ten{X}_i$ for $i = 1, ..., n$ from the conditional distribution of $\ten{X}\mid Y = y_i$ where $y_i$ is an i.i.d. sample from the standard normal distribution. The conditional distribution $\ten{X}\mid Y = y_i$ depends on the choice of the GMLM parameters $\overline{\ten{\eta}}$, $\mat{\beta}_1, ..., \mat{\beta}_r$, $\mat{\Omega}_1, ..., \mat{\Omega}_r$, and the function $\ten{F}_y$ of $y$. In all experiments we set $\overline{\ten{\eta}} = \mat{0}$. The other parameters and $\ten{F}_y$ are described per experiment. With the true GMLM parameters and $\ten{F}_y$ given, we compute the conditional tensor normal mean $\ten{\mu}_y = \ten{F}_y\mlm_{k = 1}^{r}\mat{\Omega}_k^{-1}\mat{\beta}_k$ and covariances $\mat{\Sigma}_k = \mat{\Omega}_k^{-1}$ as in \eqref{eq:tnormal_cond_params}.

We start with a $1$ dimensional linear dependence on $y$ in 1a). Then, the dependence of $y$ is via a cubic polynomial 1b-d). In 1b) reduction is full rank, in constrast to 1c) where the $\mat{\beta}_k$'s are of rank $1$, in other words, low rank regression. In 1d) we constrain the inverse covariances $\mat{\Omega}_k$ to be tri-diagonal. Both, 1c-d) are examples of building the parameter space according to \cref{thm:param-manifold}. The final tensor normal experiment 1e) is a model missspecification. The true model does \emph{not} have a Kronecker structure and the ``known'' function $\ten{F}_y$ of $y$ is missspecified as well.

\begin{itemize}
    \item[1a)] The predictors $\ten{X}$ are $2\times 3\times 5$ dimensional, that is $r = 3$. The dependence through the inverse regression model is linear specifically means that $\ten{F}_y\equiv y$ is a $1\times 1\times 1$ tensor. The true $\mat{\beta}_k$'s are all equal to $\mat{e}_1\in\mathbb{R}^{p_k}$, the first unit vector, for $k \in \{1, 2, 3\}$. The matrices $\mat{\Omega}_k = \mathrm{AR}(0.5)$ follow an auto-regression like structure. That is, the elements are given by $(\mat{\Omega}_k)_{i j} = 0.5^{|i - j|}$.
    \item[1b)] The predictors $\ten{X}$ are again $3$ dimensional with dimension $2\times 3\times 5$ which relates to the response $y$ via a qubic polynomial. This is modeled via $\ten{F}_y$ of dimension $2\times 2\times 2$ by the twice iterated outer product of the vector $(1, y)$. Element wise this reads $(\ten{F}_y)_{i j k} = y^{i + j + k - 3}$. All $\mat{\beta}_k$'s are set to $(\mat{e}_1, \mat{e}_2)\in\mathbb{R}^{p_k\times 2}$ with $\mat{e}_i$ the $i$'th unit vector and the $\mat{\Omega}_k$'s are $\mathrm{AR}(0.5)$.
    \item[1c)] Similar to 1b), except that the GMLM parameters $\mat{\beta}_k$ are rank $1$ given by
        \begin{displaymath}
            \mat{\beta}_1 = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix},\quad
            \mat{\beta}_2 = \begin{pmatrix} 1 & -1 \\ -1 & 1 \\ 1 & -1 \end{pmatrix},\quad
            \mat{\beta}_3 = \begin{pmatrix} 1 & -1 \\ -1 & 1 \\ 1 & -1 \\ -1 & 1 \\ 1 & -1 \end{pmatrix}.
        \end{displaymath}
    \item[1d)] Again like 1b). This time the true $\mat{\Omega}_k$'s, for $k = 1, 2, 3$, are tri-diagonal. Their elements are given by $(\mat{\Omega}_k)_{i j} = \delta_{0, |i - j|} + 0.5\delta_{1, |i - j|}$ with $\delta_{i, j}$ being the Kronecker delta.
    \item[1e)] For the missspecification model we let $\ten{X}\mid Y$ be multivariate normal but \emph{not} tensor normal. Let $\ten{X}$ be $5\times 5$ dimensional, $Y$ is univariate standard normal and $\mat{f}_y$ is a $4$ dimensional vector given by $\mat{f}_y = (1, \sin(y), \cos(y), \sin(y)\cos(y))$. The true vectorized reduction matrix $\mat{B}$ is $25\times 4$ consisting of the first $4$ columns of the identify. The variance-covariance matrix $\mat{\Sigma}$ is an auto-regression like structure with correlation coefficient $0.5$. Element wise $\mat{B}_{i j} = \delta_{i j}$ and $\mat{\Sigma}_{i j} = 0.5^{|i - j|}$. Both, $\mat{B}$ and $\mat{\Omega} = \mat{\Sigma}^{-1}$ violate the Kronecker product assumptions \eqref{eq:eta1} and \eqref{eq:eta2} of the GMLM model. Then, we set
    \begin{displaymath}
        \vec{\ten{X}}\mid (Y = y) = \mat{B}\mat{f}_y + \mathcal{N}_{25}(\mat{0}, \mat{\Sigma}).
    \end{displaymath}
    Furthermore, we fit the model with the wrong ``known'' function $\ten{F}_y$. We set $\ten{F}_y$ to be a $2\times 2$ matrix with a quadratic linkage via elements given by $(\ten{F}_y)_{i j} = y^{|i - j|}$.
\end{itemize}



\begin{figure}
    \centering
    \includegraphics[width = \textwidth]{plots/sim-normal.pdf}
    \caption{\label{fig:sim-normal}asknclknasknc}
\end{figure}


\begin{figure}[ht!]
    \centering
    \includegraphics[width = \textwidth]{plots/sim-tsir.pdf}
    \caption{\label{fig:sim-tsir}Simulation to investiage the unexpected failure of TSIR in simulation 1c.}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Ising Model}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}
    \centering
    \includegraphics[width = \textwidth]{plots/sim-ising.pdf}
    \caption{\label{fig:sim-ising}asknclknasknc}
\end{figure}

\begin{table}
    \begin{tabular}{c | ccc ccc c}
        $n$ &             GMLM  &         PCA &       HOPCA &       LPCA  &       CLPCA &        TSIR &       MGCCA \\
        \hline
        100 & {\bf 0.34} (0.14) & 0.90 (0.04) & 0.90 (0.05) & 0.94 (0.09) & 0.91 (0.03) & 0.48 (0.19) & 0.55 (0.13) \\
        200 & {\bf 0.25} (0.11) & 0.90 (0.03) & 0.90 (0.03) & 0.96 (0.07) & 0.91 (0.02) & 0.38 (0.16) & 0.53 (0.10) \\
        300 & {\bf 0.20} (0.09) & 0.89 (0.02) & 0.89 (0.02) & 0.97 (0.06) & 0.91 (0.02) & 0.29 (0.13) & 0.51 (0.11) \\
        500 & {\bf 0.16} (0.07) & 0.90 (0.02) & 0.90 (0.02) & 0.98 (0.01) & 0.91 (0.01) & 0.23 (0.10) & 0.50 (0.08) \\
        750 & {\bf 0.13} (0.05) & 0.90 (0.01) & 0.90 (0.01) & 0.98 (0.02) & 0.91 (0.01) & 0.23 (0.08) & 0.53 (0.06)
    \end{tabular}
    \caption{\label{tab:sim-ising}xyz uvw}
\end{table}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Data Analysis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section be perform two \todo{realy two!} applications of the GMLM model on real data. First example is the tensor normal model applied to EED data\todo{do this!}. Next, we perform a prove of concept data analysis example for chess. The main purpose of choosing chess is two fold. First, we can ilustrate an explicit use case for the (till now ignored) linear constraint matrix $\mat{T}_2$. Second, its a personal interest of one of authors.\todo{???}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{EEG}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Chess}\label{sec:chess}
The data set is provided by the \citetitle{lichess-database}\footnote{\fullcite{lichess-database}}. We downloaded November of 2023 consisting of more than $92$ million games. We removed all games without Stockfish\footnote{\fullcite{stockfish}} position evaluations. Those take the role of the response $Y$ and correspond to a winning probability from whites point of few. Positive scores are good for white and negative scores indicate an advantage for black. We ignore all highlty unballanced positions, which we set to be positions with absolute score above $5$. We also remove all positions with a mate score (one side can force check mate). Finally, we only considure positions with white to move. This leads to a final data set of roughly $190$ million positions, including duplicates.

A chess position is encoded as a 3D binary tensor of dimension $8\times 8\times 12$ giving the predictors $\ten{X}$, see \cref{fig:fen2tensor}. The first two axis encode the squares of a chess board, which is a $8\times 8$ grid. The third axis encodes chess pieces. The $12$ pieces derive from the $6$ types of pieces, namely pawns (\pawn), knights (\knight), bishops (\bishop), queens (\queen) and kings (\king) of two colors, black and white.

\begin{figure}[hp!]
    \centering
    \includegraphics[width = \textwidth]{images/fen2tensor.pdf}
    \caption{\label{fig:fen2tensor}The chess start position and its 3D binary tensor representation, empty entries are $0$.}
\end{figure}

Now, we assume that $(\ten{X}, y)$ follows the Ising GMLM model \cref{sec:ising_estimation}. The relation to the scores $y$ is modeled via a cubic relation by the $2\times 2\times 2$ dimensional tensor $\ten{F}_y$ with elements $(\ten{F}_y)_{i j k} = y^{i + j + k - 3}$.

Due to the rules of chess, given any encoded legal position $\ten{X}$, there are some elements which are never set. This are the elements corresponding to pawns on the 1st and 8th rank. This implies that the matrix $\mat{T}_2$ in \eqref{eq:t-stat} under model \eqref{eq:quadratic-exp-fam} can \emph{not} be the identity as, for simplicity, assumed in \cref{sec:ising_estimation}. The addaptation in this case is simple. We only need to enforce the corresponding diagonal elements in $\mat{\Omega}$ to be zero, which is a linear operation. The only difference in the estimation procedure is to use $\ten{G}_2$ in \cref{thm:grad} for gradient computations. This can be extended to incorporate additional constraints into the model derived from the game of chess. First, all interactions of pieces with pawns on the 1st or 8th rank are impossible. Next, there is always exactly one king per color on the board and they can not be on neighboring squares. Finally, only one piece can occupy a square. All those conditions can be writen as a linear constraint on $\mat{\Omega}$ by setting the corresponding entries in $\mat{\Omega}$ to zero.

By the raw scale of the data, millions of observations, it is computationaly infeasible to compute the gradients on the entire data set. Simply using a computationaly managable subset is not an option. The problem is that the Ising model is of dimension $8\cdot 8\cdot 12 = 768$ which requires an absurd amount of data to get a reasonable picture of the interaction effects. The solution is to switch from a classic gradient based optimization to a stochastic version. This means that every gradient update uses a new random subset of the entire data set. Therefore, we draw independent random samples of the data. The independence of samples derived from the independence of games, and every sample is drawn from a different game.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Tensor Calculus and Multi Linear Algebra}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{displaymath}
    (\ten{A}\circ\ten{B})\mlm_{k = 1}^{r + s} \mat{C}_k
    =
    \Bigl(\ten{A}\mlm_{k = 1}^r \mat{C}_k\Bigr)\circ\Bigl(\ten{B}\mlm_{l = 1}^s \mat{C}_{l + r}\Bigr)
\end{displaymath}
Using $\K(\ten{A}\circ\ten{B}) = \ten{A}\otimes\ten{B}$ gives
\begin{displaymath}
    \K\Bigl((\ten{A}\circ\ten{B})\mlm_{k = 1}^{r + s} \mat{C}_k\Bigr)
    =
    \Bigl(\ten{A}\mlm_{k = 1}^r \mat{C}_k\Bigr)\otimes\Bigl(\ten{B}\mlm_{l = 1}^s \mat{C}_{l + r}\Bigr)
\end{displaymath}
A generalization of the well known identity $\vec(\mat{A}\mat{B}\mat{C}) = (\t{\mat{C}}\otimes\mat{A})\vec{\mat{B}}$ is given by
\begin{displaymath}
    \Bigl(\ten{A}\mlm_{k = 1}^r \mat{B}_k \Bigr)_{(\mat{i}, \mat{j})}
    =
    \Bigl( \bigotimes_{k = \#\mat{i}}^{1}\mat{B}_{\mat{i}_k} \Bigr)
        \ten{A}_{(\mat{i}, \mat{j})}
            \Bigl( \bigotimes_{l = \#\mat{j}}^{1}\t{\mat{B}_{\mat{j}_l}} \Bigr)
\end{displaymath}
with the special case
\begin{displaymath}
    \vec\Bigl(\ten{A}\mlm_{k = 1}^r \mat{B}_k\Bigr)
    =
    \Bigl(\bigotimes_{k = r}^{1}\mat{B}_k\Bigr)\vec{\ten{A}}
\end{displaymath}

Furthermore, we have
\begin{displaymath}
    (\ten{A}\otimes\ten{B})\mlm_{k = 1}^{r}\t{(\vec\mat{C}_k)}
    =
    \Bigl\langle \ten{A}\mlm_{k = 1}^{r} \mat{C}_k, \ten{B} \Bigr\rangle
    =
    \Bigl\langle \ten{A}, \ten{B}\mlm_{k = 1}^{r} \t{\mat{C}_k} \Bigr\rangle
    =
    \t{(\vec{\ten{B}})}\Bigl(\bigotimes_{k = r}^{1}\mat{C}_k\Bigr)\vec{\ten{A}}
\end{displaymath}
as well as for any tensor $\ten{A}$ of even order $2 r$ and matching square matrices $\mat{B}_k$ holds
\begin{displaymath}
    \K(\ten{A})\mlm_{k = 1}^{r}\t{(\vec\mat{B}_k)}
    =
    \t{(\vec{\ten{A}})}\vec\Bigl(\bigotimes_{k = r}^{1}\t{\mat{B}_k}\Bigr)
\end{displaymath}


% \begin{lemma}\label{thm:kron-perm}
%     Given $r$ matrices $\mat{A}_k$ of dimension $p_j\times q_j$ for $k = 1, \ldots, r$, then there exists a unique permutation matrix $\mat{S}_{\mat{p}, \mat{q}}$ such that
%     \begin{equation}\label{eq:kron-to-outer-perm}
%         \vec\bigkron_{k = r}^{1}\mat{A}_j = \mat{S}_{\mat{p}, \mat{q}}\vec\bigouter_{k = 1}^{r}\mat{A}_k.
%     \end{equation}
%     The permutation $\mat{S}_{\mat{p}, \mat{q}}$ with indices $\mat{p} = (p_1, \ldots, p_r)$ and $\mat{q} = (q_1, \ldots, q_r)$ is the matrix-matrix product of $r - 1$ permutation matrices given by
%     \begin{multline}\label{eq:S_pq}
%         \mat{S}_{\mat{p}, \mat{q}} =
%             \Bigl[ \mat{I}_1\otimes \Bigl( \mat{I}_{\prod_{k = r}^{2}q_k}\otimes\mat{K}_{q_1, \prod_{k = r}^{2}p_k}\otimes I_{p_1} \Bigr)\Bigr] \\
%             \Bigl[ \mat{I}_{p_1 q_1}\otimes \Bigl( \mat{I}_{\prod_{k = r}^{3}q_k}\otimes\mat{K}_{q_2, \prod_{k = r}^{3}p_k}\otimes I_{p_2} \Bigr) \Bigr]
%             \cdots
%             \Bigl[ \mat{I}_{\prod_{k = 1}^{r - 2}p_k q_k}\otimes \Bigl( \mat{I}_{q_r}\otimes\mat{K}_{q_{r - 1}, p_r}\otimes I_{p_{r - 1}} \Bigr) \Bigr]
%     \end{multline}
%     where $\mat{K}_{p, q}$ is the \emph{commutation matrix} from \textcite[Ch.~11]{MatrixAlgebra-AbadirMagnus2005}, that is the permutation such that $\vec{\t{\mat{A}}} = \mat{K}_{p, q}\vec{\mat{A}}$ for every $p\times q$ dimensional matrix $\mat{A}$.
% \end{lemma}
% \begin{proof}
%     \textcite[Lemma~7]{SymMatandJacobians-MagnusNeudecker1986} states that
%     \begin{align*}
%         \vec(\mat{A}_2\otimes\mat{A}_1)
%             &= (\mat{I}_{q_2}\otimes\mat{K}_{q_1, p_2}\otimes\mat{I}_{p_1})(\vec{\mat{A}_2}\otimes\vec{\mat{A}_1}) \\
%             &= (\mat{I}_{q_2}\otimes\mat{K}_{q_1, p_2}\otimes\mat{I}_{p_1})\vec(\mat{A}_1\circ \mat{A}_2).
%     \end{align*}
%     This proves the statement for $r = 2$. The general statement for $r > 2$ follows via induction using \textcite[Lemma~7]{SymMatandJacobians-MagnusNeudecker1986} in conjunction with $\vec(\mat{C}\mat{a}\t{\mat{b}}) = (\mat{I}_{\dim(\mat{b})}\otimes\mat{C})\vec(\mat{a}\t{\mat{b}})$.
% \end{proof}
\begin{lemma}\label{thm:kron-perm}
    Given $r \geq 2$ matrices $\mat{A}_k$ of dimension $p_j\times q_j$ for $k = 1, \ldots, r$, then there exists a unique permutation matrix $\mat{S}_{\mat{p}, \mat{q}}$ such that
    \begin{equation}\label{eq:kron-to-outer-perm}
        \vec\bigkron_{k = r}^{1}\mat{A}_k = \mat{S}_{\mat{p}, \mat{q}}\vec\bigouter_{k = 1}^{r}\mat{A}_k.
    \end{equation}
    The permutation $\mat{S}_{\mat{p}, \mat{q}}$ with indices $\mat{p} = (p_1, \ldots, p_r)$ and $\mat{q} = (q_1, \ldots, q_r)$ is defined recursively as
    \begin{equation}\label{eq:S_pq}
        \mat{S}_{\mat{p}, \mat{q}} = \mat{S}_{\bigl( \prod_{k = 1}^{r - 1}p_k, p_r \bigr), \bigl( \prod_{k = 1}^{r - 1}q_k, q_r \bigr)} \bigl(\mat{I}_{p_r q_r}\otimes\mat{S}_{(p_1, \ldots, p_{r-1}), (q_1, \ldots, q_{r-1})}\bigr)
    \end{equation}
    with initial value
    \begin{displaymath}
        \mat{S}_{(p_1, p_2), (q_1, q_2)} = \mat{I}_{q_2}\otimes\mat{K}_{q_1, p_2}\otimes\mat{I}_{p_1}
    \end{displaymath}
    where $\mat{K}_{p, q}$ is the \emph{commutation matrix} from \textcite[Ch.~11]{MatrixAlgebra-AbadirMagnus2005}, that is the permutation such that $\vec{\t{\mat{A}}} = \mat{K}_{p, q}\vec{\mat{A}}$ for every $p\times q$ dimensional matrix $\mat{A}$.
\end{lemma}
\begin{proof}
    \textcite[Lemma~7]{SymMatandJacobians-MagnusNeudecker1986} states that
    \begin{align}
        \vec(\mat{A}_2\otimes\mat{A}_1)
            &= (\mat{I}_{q_2}\otimes\mat{K}_{q_1, p_2}\otimes\mat{I}_{p_1})(\vec{\mat{A}_2}\otimes\vec{\mat{A}_1}) \label{eq:MagnusNeudecker1986-vec-kron-identity} \\
            &= (\mat{I}_{q_2}\otimes\mat{K}_{q_1, p_2}\otimes\mat{I}_{p_1})\vec(\mat{A}_1\circ \mat{A}_2). \nonumber
    \end{align}
    This proves the statement for $r = 2$. The general statement for $r > 2$ follows via induction. Assuming \eqref{eq:kron-to-outer-perm} holds for $r - 1$, the induction step is then;
    \begin{multline*}
        \vec{\bigkron_{k = r}^{1}}\mat{A}_k
        = \vec\Bigl(\mat{A}_r\otimes\bigkron_{k = r - 1}^{1}\mat{A}_k\Bigr)
        \overset{\eqref{eq:MagnusNeudecker1986-vec-kron-identity}}{=} \Bigl( \mat{I}_{q_r}\otimes\mat{K}_{\prod_{k = 1}^{r - 1}q_k, p_r}\otimes\mat{I}_{\prod_{k = 1}^{r - 1}p_k} \Bigr)\vec\Bigl((\vec\mat{A}_r)\otimes\vec\bigkron_{k = r - 1}^{1}\mat{A}_k\Bigr) \\
        = \mat{S}_{\bigl( \prod_{k = 1}^{r - 1}p_k, p_r \bigr), \bigl( \prod_{k = 1}^{r - 1}q_k, q_r \bigr)}\vec\Bigl[\Bigl(\vec\bigkron_{k = r - 1}^{1}\mat{A}_k\Bigr)\t{(\vec\mat{A}_r)}\Bigr] \\
        \overset{\eqref{eq:kron-to-outer-perm}}{=} \mat{S}_{\bigl( \prod_{k = 1}^{r - 1}p_k, p_r \bigr), \bigl( \prod_{k = 1}^{r - 1}q_k, q_r \bigr)}\vec\Bigl[\mat{S}_{(p_1, \ldots, p_{r-1}), (q_1, \ldots, q_{r-1})}\Bigl(\vec\bigouter_{k = 1}^{r - 1}\mat{A}_k\Bigr)\t{(\vec\mat{A}_r)}\Bigr] \\
        \overset{(a)}{=} \mat{S}_{\bigl( \prod_{k = 1}^{r - 1}p_k, p_r \bigr), \bigl( \prod_{k = 1}^{r - 1}q_k, q_r \bigr)} \bigl(\mat{I}_{p_r q_r}\otimes\mat{S}_{(p_1, \ldots, p_{r-1}), (q_1, \ldots, q_{r-1})}\bigr)\vec\Bigl[\Bigl(\vec\bigouter_{k = 1}^{r - 1}\mat{A}_k\Bigr)\t{(\vec\mat{A}_r)}\Bigr] \\
        = \mat{S}_{\mat{p}, \mat{q}}\vec\bigouter_{k = 1}^{r}\mat{A}_k.
    \end{multline*}
    Equality $(a)$ uses the relation $\vec(\mat{C}\mat{a}\t{\mat{b}}) = (\mat{I}_{\dim(\mat{b})}\otimes\mat{C})\vec(\mat{a}\t{\mat{b}})$ for a matrix $\mat{C}$ and vectors $\mat{a}, \mat{b}$.
\end{proof}

% \begin{remark}
%     \todo{simplification of $\mat{S}$ if all dimensions are equal, that is if $p_k = p_j$ and $p_k = q_k$ for all $k, j$?!}
% \end{remark}

\begin{remark}
    The permutation matrix $\mat{K}_{p, q}$ represents a perfect outer $p$-shuffle of $p q$ elements.
\end{remark}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proofs}\label{app:B}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}[Proof of \cref{thm:sdr}]\label{proof:sdr}
    A direct implication of \textcite[Theorem~1]{sdr-BuraDuarteForzani2016} is that, under the exponential family \eqref{eq:quadratic-exp-fam} with natural statistic \eqref{eq:t-stat},
    \begin{displaymath}
        \t{\mat{\alpha}}(\mat{t}(\ten{X}) - \E\mat{t}(\ten{X}))
    \end{displaymath}
    is a sufficient reduction, where $\mat{\alpha}\in\mathbb{R}^{(p + d)\times q}$ with $\Span(\mat{\alpha}) = \Span(\{\mat{\eta}_y - \E_{Y}\mat{\eta}_Y : y\in\mathcal{S}_Y\})$. Since $\E_Y\ten{F}_Y = 0$, $\E_Y\mat{\eta}_{1 Y} = \E[\vec\overline{\ten{\eta}} - \mat{B}\vec\ten{F}_Y] = \vec\overline{\ten{\eta}}$. Thus,  
    \begin{displaymath}
        \mat{\eta}_y - \E_{Y}\mat{\eta}_Y = \begin{pmatrix}
            \mat{\eta}_{1 y} - \E_{Y}\mat{\eta}_{1 Y} \\
            \mat{\eta}_{2} - \E_{Y}\mat{\eta}_{2}
        \end{pmatrix} = \begin{pmatrix}
            \mat{B}\vec\ten{F}_y \\
            \mat{0}
        \end{pmatrix}.
    \end{displaymath}
as $\mat{\eta}_{2}$ does not depend on $y$. 
The set  $\{ \vec{\ten{F}_y} : y\in\mathcal{S}_y \}$ is a subset of $\mathbb{R}^q$. Therefore,
    \begin{displaymath}
        \Span\left(\{\mat{\eta}_y - \E_{Y}\mat{\eta}_Y : y\in\mathcal{S}_Y\}\right) = \Span\left(\left\{\begin{pmatrix}
            \mat{B}\vec\ten{F}_Y \\ \mat{0}
        \end{pmatrix} : y\in\mathcal{S}_Y \right\}\right)
        \subseteq
        \Span\begin{pmatrix}
            \mat{B} \\ \mat{0}
        \end{pmatrix},
    \end{displaymath}
which obtains that 
    \begin{displaymath}
        \t{\begin{pmatrix}
            \mat{B} \\ \mat{0}
        \end{pmatrix}}(\mat{t}(\ten{X}) - \E\mat{t}(\ten{X}))
        =
        \t{\mat{B}}\vec(\ten{X} - \E\ten{X})
        = \vec\Bigl(\ten{F}_y\mlm_{k = 1}^{r}\mat{\beta}_k\Bigr)
    \end{displaymath}
is also a sufficient reduction, though not necessarily minimal,     using $\mat{B} = \bigkron_{k = 1}^{r}\mat{\beta}_k$. When the exponential family is full rank, which in our setting amounts to all $\mat{\beta}_j$ being full rank matrices, $j=1,\ldots,r$, then \textcite[Thm~1]{sdr-BuraDuarteForzani2016} also obtains the minimality of the reduction.  
\end{proof}

\todo{check proof of Thm 2}
\begin{proof}[Proof of \cref{thm:grad}]\label{proof:grad}
    We first note that for any exponential family with density \eqref{eq:quad-density} the term $b(\mat{\eta}_{y_i})$ differentiated with respect to the natural parameter $\mat{\eta}_{y_i}$ is the expectation of the statistic $\mat{t}(\ten{X})$ given $Y = y_i$. In our case we get $\nabla_{\mat{\eta}_{y_i}}b = (\nabla_{\mat{\eta}_{1{y_i}}}b, \nabla_{\mat{\eta}_2}b)$ with components
    \begin{displaymath}
        \nabla_{\mat{\eta}_{1{y_i}}}b
            = \E[\mat{t}_1(\ten{X})\mid Y = y_i]
            = \vec\E[\ten{X}\mid Y = y_i]
            = \vec\ten{g}_1(\mat{\eta}_{y_i})
    \end{displaymath}
    and
    \begin{align*}
        \nabla_{\mat{\eta}_{2}}b
           &= \E[\mat{t}_2(\ten{X})\mid Y = y_i]
            = \E[\mat{T}_2\vech((\vec\ten{X})\t{(\vec\ten{X})})\mid Y = y_i] \\
           &= \E[\mat{T}_2\pinv{\mat{D}_p}\vec(\ten{X}\circ\ten{X})\mid Y = y_i]
            = \mat{T}_2\pinv{\mat{D}_p}\vec\ten{g}_2(\mat{\eta}_{y_i}).
    \end{align*}
    The gradients are related to their derivatives by transposition, $\nabla_{\mat{\eta}_{1{y_i}}}b = \t{\D b(\mat{\eta}_{1{y_i}})}$ and $\nabla_{\mat{\eta}_2}b = \t{\D b(\mat{\eta}_2)}$.
    Next we provide the differentials of the natural parameter components from \eqref{eq:eta1} and \eqref{eq:eta2} in a quite direct form ,without any further ``simplifications'', because the down-stream computations won't benefit from reexpressing the following
    \todo{in terms of $m_{\mat{\theta}}$, meaning without the sum, makes it a bit nicer!}
    \begin{align*}
        \d\mat{\eta}_{1{y_i}}(\overline{\ten{\eta}})
            &= \d\vec{\overline{\ten{\eta}}}, \\
        \d\mat{\eta}_{1{y_i}}(\mat{\beta}_j)
            &= \vec\Bigl( \ten{F}_{y_i}\mlm_{\substack{k = 1\\k\neq j}}^{r}\mat{\beta}_k\times_j\d\mat{\beta}_j \Bigr), \\
        \d\mat{\eta}_2(\mat{\Omega}_j)
            &= \t{(\pinv{(\mat{T}_2\pinv{\mat{D}_p})})}\vec(c\,\d\mat{\Omega}) \\
            &= c\t{(\pinv{(\mat{T}_2\pinv{\mat{D}_p})})}\vec\Bigl(\,\bigkron_{k = r}^{j + 1}\mat{\Omega}_k\otimes\d\mat{\Omega}_j\otimes\bigkron_{l = j - 1}^{1}\mat{\Omega}_l \Bigr).
    \end{align*}
    All other combinations, namely $\d\mat{\eta}_{1{y_i}}(\mat{\Omega}_j)$, $\d\mat{\eta}_2(\overline{\ten{\eta}})$ and $\d\mat{\eta}_2(\mat{\beta}_j)$, are zero.
    Continuing with the partial differentials of $l_n$ from \eqref{eq:log-likelihood}
    \begin{multline*}
        \d l_n(\overline{\ten{\eta}})
            = \sum_{i = 1}^{n} (\langle \d\overline{\ten{\eta}}, \ten{X}_i \rangle - \D b(\mat{\eta}_{1{y_i}})\d\mat{\eta}_{1{y_i}}(\overline{\ten{\eta}}))
            = \sum_{i = 1}^{n} \t{(\vec{\ten{X}_i} - \vec\ten{g}_1(\mat{\eta}_{y_i}))}\d\vec{\overline{\ten{\eta}}} \\
            = \t{(\d\vec{\overline{\ten{\eta}}})}\vec\sum_{i = 1}^{n} (\ten{X}_i - \ten{g}_1(\mat{\eta}_{y_i})).
    \end{multline*}
    For every $j = 1, ..., r$ we get the differentials
    \begin{multline*}
        \d l_n(\mat{\beta}_j)
             = \sum_{i = 1}^{n} \biggl(\Bigl\langle \ten{F}_{y_i}\mlm_{\substack{k = 1\\k\neq j}}^{r}\mat{\beta}_k\times_j\d\mat{\beta}_j, \ten{X}_i \Bigr\rangle - \D b(\mat{\eta}_{1{y_i}})\d\mat{\eta}_{1{y_i}}(\mat{\beta}_j)\biggr)
             = \sum_{i = 1}^{n} \Bigl\langle \ten{F}_{y_i}\mlm_{\substack{k = 1\\k\neq j}}^{r}\mat{\beta}_k\times_j\d\mat{\beta}_j, \ten{X}_i - \ten{g}_1(\mat{\eta}_{y_i}) \Bigr\rangle \\
             = \sum_{i = 1}^{n} \tr\biggl( \d\mat{\beta}_j\Bigl(\ten{F}_{y_i}\mlm_{\substack{k = 1\\k\neq j}}^{r}\mat{\beta}_k\Bigr)_{(j)} \t{(\ten{X}_i - \ten{g}_1(\mat{\eta}_{y_i}))_{(j)}} \biggr)
             = \t{(\d\vec{\mat{\beta}_j})}\vec\sum_{i = 1}^{n} (\ten{X}_i - \ten{g}_1(\mat{\eta}_{y_i}))_{(j)} \t{\Bigl(\ten{F}_{y_i}\mlm_{\substack{k = 1\\k\neq j}}^{r}\mat{\beta}_k\Bigr)_{(j)}}
    \end{multline*}
    as well as
    \begin{multline*}
        \d l_n(\mat{\Omega}_j)
            = \sum_{i = 1}^{n} \biggl( c\Bigl\langle \ten{X}_i\mlm_{\substack{k = 1\\k\neq j}}^{r}\mat{\Omega}_k\times_j\d\mat{\Omega}_j, \ten{X}_i \Bigr\rangle - \D b(\mat{\eta}_2)\d\mat{\eta}_2(\mat{\Omega}_j) \biggr) \\
            = c\sum_{i = 1}^{n} \biggl( \Bigl\langle \ten{X}_i\mlm_{\substack{k = 1\\k\neq j}}^{r}\mat{\Omega}_k\times_j\d\mat{\Omega}_j, \ten{X}_i \Bigr\rangle - \t{(\mat{T}_2\pinv{\mat{D}_p}\vec\ten{g}_2(\mat{\eta}_{y_i}))}\t{(\pinv{(\mat{T}_2\pinv{\mat{D}_p})})}\vec\Bigl(\,\bigkron_{k = r}^{j + 1}\mat{\Omega}_k\otimes\d\mat{\Omega}_j\otimes\bigkron_{l = j - 1}^{1}\mat{\Omega}_l \Bigr) \biggr) \\
            = c\sum_{i = 1}^{n} \biggl( \Bigl\langle \ten{X}_i\mlm_{\substack{k = 1\\k\neq j}}^{r}\mat{\Omega}_k\times_j\d\mat{\Omega}_j, \ten{X}_i \Bigr\rangle - \t{(\vec\ten{G}_2(\mat{\eta}_{y_i}))}\vec\Bigl(\,\bigkron_{k = r}^{j + 1}\mat{\Omega}_k\otimes\d\mat{\Omega}_j\otimes\bigkron_{l = j - 1}^{1}\mat{\Omega}_l \Bigr) \biggr) \\
            = c\sum_{i = 1}^{n} \biggl( \t{\vec(\ten{X}_i\circ\ten{X}_i - \ten{G}_2(\mat{\eta}_{y_i}))}\vec\Bigl(\,\bigkron_{k = r}^{j + 1}\mat{\Omega}_k\otimes\d\mat{\Omega}_j\otimes\bigkron_{l = j - 1}^{1}\mat{\Omega}_l \Bigr) \biggr) \\
            = c\sum_{i = 1}^{n} \K(\ten{X}_i\circ\ten{X}_i - \ten{G}_2(\mat{\eta}_{y_i}))\mlm_{\substack{k = 1\\k\neq j}}^{r}\t{(\vec{\mat{\Omega}_k})}\times_j\t{(\d\vec{\mat{\Omega}_j})} \\
            = c\t{(\d\vec{\mat{\Omega}_j})}\sum_{i = 1}^{n} \Bigl((\ten{X}_i\otimes\ten{X}_i - \K(\ten{G}_2(\mat{\eta}_{y_i})))\mlm_{\substack{k = 1\\k\neq j}}^{r}\t{(\vec{\mat{\Omega}_k})}\Bigr)_{(j)} \\
            = c\t{(\d\vec{\mat{\Omega}_j})}\vec\sum_{i = 1}^{n} (\ten{X}_i\otimes\ten{X}_i - \K(\ten{G}_2(\mat{\eta}_{y_i})))\mlm_{\substack{k = 1\\k\neq j}}^{r}\t{(\vec{\mat{\Omega}_k})}
    \end{multline*}
    Now, applying the identity $\d \ten{A}(\ten{B}) = \t{(\d\vec{\ten{B}})}\nabla_{\ten{B}}\ten{A}$ gives the required partial gradients.

    Finally, if $\mat{T}_2$ is the identify matrix, then
    \begin{displaymath}
        \vec{\ten{G}_2(\mat{\eta}_y)} = \pinv{(\mat{T}_2\pinv{\mat{D}_p})}\mat{T}_2\pinv{\mat{D}_p}\vec{\ten{g}_2(\mat{\eta}_y)}
            = \mat{D}_p\pinv{\mat{D}_p}\vec{\ten{g}_2(\mat{\eta}_y)}
            = \vec{\ten{g}_2(\mat{\eta}_y)}
    \end{displaymath}
    where the last equality holds because $\mat{N}_p = \mat{D}_p\pinv{\mat{D}_p}$ is the symmetrizer matrix from \textcite[Ch. 11]{MatrixAlgebra-AbadirMagnus2005}. For the symmetrizer matrix $\mat{N}_p$ holds $\mat{N}_p\vec{\mat{A}} = \vec{\mat{A}}$ if $\mat{A} = \t{\mat{A}}$, while
    \begin{displaymath}
        \vec{\ten{g}_2(\mat{\eta}_y)} = \vec\E[\ten{X}\circ\ten{X}\mid Y = y] = \vec\E[(\vec{\ten{X}})\t{(\vec{\ten{X}})}\mid Y = y]
    \end{displaymath}
    is the vectorization of a symmetric matrix.
\end{proof}

\begin{proof}[Proof of \cref{thm:kron-manifolds}]\label{proof:kron-manifolds}
    We start by considering the first case and assume that $\manifold{B}$ is spherical with radius $1$ w.l.o.g. We equip $\manifold{K} = \{ \mat{A}\otimes \mat{B} : \mat{A}\in\manifold{A}, \mat{B}\in\manifold{B} \}\subset\mathbb{R}^{p_1 p_2\times q_1 q_2}$ with the subspace topology \efi{cite{?}}. Define the hemispheres $H_i^{+} = \{ \mat{B}\in\manifold{B} : (\vec{\mat{B}})_i > 0 \}$ and $H_i^{-} = \{ \mat{B}\in\manifold{B} : (\vec{\mat{B}})_i < 0 \}$ for $i = 1, ..., p_2 q_2$. The hemispheres are an open cover of $\manifold{B}$ with respect to the subspace topology. Define for every $H_i^{\pm}$, where $\pm$ is a placeholder for ether $+$ or $-$, the function
    \begin{displaymath}
        f_{H_i^{\pm}} : \manifold{A}\times H_i^{\pm}\to\mathbb{R}^{p_1 p_2\times q_1 q_2}
            : (\mat{A}, \mat{B})\mapsto \mat{A}\otimes \mat{B}
    \end{displaymath}
    which is smooth. With the spherical property of $\manifold{B}$ the relation $\|\mat{A}\otimes \mat{B}\|_F = \|\mat{A}\|_F$ for all $\mat{A}\otimes \mat{B}\in\manifold{K}$ ensures that the function $f_{H_i^{\pm}}$, constrained to its image, is bijective with inverse function (identifying $\mathbb{R}^{p\times q}$ with $\mathbb{R}^{p q}$) given by
    \begin{displaymath}
        f_{H_i^{\pm}}^{-1} : f_{H_i^{\pm}}(\manifold{A}\times H_i^{\pm})\to\manifold{A}\times H_i^{\pm}
            : \mat{C}\mapsto \left(\pm\frac{\|\mat{C}\|_F}{\|\mat{R}(\mat{C})\mat{e}_i\|_2}\mat{R}(\mat{C})\mat{e}_i, \pm\frac{1}{\|\mat{C}\|_F\|\mat{R}(\mat{C})\mat{e}_i\|_2}\mat{R}(\mat{C})\t{\mat{R}(\mat{C})}\mat{e}_i\right)
    \end{displaymath}
    where $\pm$ is $+$ for a ``positive'' hemisphere $H_i^{+}$ and $-$ otherwise, $\mat{e}_i\in\mathbb{R}^{p_2 q_2}$ is the $i$th unit vector and $\mat{R}(\mat{C})$ is a ``reshaping'' permutation \footnote{Relating to $\K$ the operation $\mat{R}$ is basically its inverse as $\K(\mat{A}\circ\mat{B}) = \mat{A}\otimes\mat{B}$ with a mismatch in the shapes only.} which acts on Kronecker products as $\mat{R}(\mat{A}\otimes \mat{B}) = (\vec{\mat{A}})\t{(\vec{\mat{B}})}$. This makes $f_{H_i^{\pm}}^{-1}$ a combination of smooth functions ($\mat{0}$ is excluded from $\manifold{A}, \manifold{B}$ guarding against division by zero) and as such it is also smooth. This ensures that $f_{H_i^{\pm}} : \manifold{A}\times {H_i^{\pm}}\to f_{H_i^{\pm}}(\manifold{A}\times {H_i^{\pm}})$ is a diffeomorphism.

    Next, we construct an atlas\footnote{A collection of charts $\{ \varphi_i : i\in I \}$ with index set $I$ of a manifold $\manifold{A}$ is called an \emph{atlas} if the pre-images of the charts $\varphi_i$ cover the entire manifold $\manifold{A}$.} for $\manifold{K}$ which is equipped with the subspace topology. Let $(\varphi_j, U_j)_{j\in J}$ be a atlas of $\manifold{A}\times\manifold{B}$. Such an atlas exists and admits a unique smooth structure as both $\manifold{A}, \manifold{B}$ are embedded manifolds from which we take the product manifold. The images of the coordinate domains $f_H(U_j)$ are open in $\manifold{K}$, since $f_H$ is a diffeomorphism, with the corresponding coordinate maps
    \begin{displaymath}
        \phi_{H_i^{\pm},j} : f_{H_i^{\pm}}(U_j)\to \varphi_j(U_j)
            : \mat{C}\mapsto \varphi_j(f_{H_i^{\pm}}^{-1}(\mat{C})).
    \end{displaymath}
    By construction the set $\{ \phi_{H_i^{\pm},j} : i = 1, ..., p_2 q_2, \pm\in\{+, -\}, j\in J \}$ is an atlas if the charts are compatible. This means we need to check if the transition maps are diffeomorphisms, let $(\phi_{H, j}, V_j), (\phi_{\widetilde{H}, k}, V_k)$ be two arbitrary charts from our atlas, then the transition map $\phi_{\widetilde{H}, k}\circ\phi_{H,j}^{-1}:\phi_{H,j}^{-1}(V_j\cap V_k)\to\phi_{\widetilde{H},k}^{-1}(V_j\cap V_k)$ has the form
    \begin{displaymath}
        \phi_{\widetilde{H}, k}\circ\phi_{H,j}^{-1}
            = \varphi_k\circ f_{\widetilde{H}}^{-1}\circ f_{H}\circ\varphi_{j}^{-1}
            = \varphi_k\circ (\pm\mathrm{id})\circ\varphi_{j}^{-1}
    \end{displaymath}
    where $\pm$ depends on $H, \widetilde{H}$ being of the same ``sign'' and $\mathrm{id}$ is the identity. We conclude that the charts are compatible, which makes the constructed set of charts an atlas. With that we have shown the topological manifold $\manifold{K}$ with the subspace topology admit a smooth atlas that makes it an embedded smooth manifold with dimension equal to the dimension of the product topology $\manifold{A}\times\manifold{B}$; that is, $d = \dim\manifold{A} + \dim\manifold{B}$.

    It remains to show that the cone condition also admits a smooth manifold. $\manifold{K} = \{ \mat{A}\otimes \mat{B} : \mat{A}\in\manifold{A}, \mat{B}\in\widetilde{B} \}$, where $\widetilde{B} = \{ \mat{B}\in\manifold{B} : \|\mat{B}\|_F = 1 \}$, holds if both $\manifold{A}, \manifold{B}$ are cones. Since $g:\manifold{B}\to\mathbb{R}:\mat{B}\mapsto \|\mat{B}\|_F$ is continuous on $\manifold{B}$ with full rank $1$ everywhere,  $\widetilde{\manifold{B}} = g^{-1}(1)$ is a $\dim{\manifold{B}} - 1$ dimensional embedded submanifold of $\manifold{B}$. An application of the spherical case proves the cone case.
\end{proof}

\begin{proof}[Proof of \cref{thm:param-manifold}]
    An application of \cref{thm:kron-manifold-tangent-space} ensures that $\manifold{K}_{\mat{B}}$ and $\manifold{K}_{\mat{\Omega}}$ are embedded submanifolds.

    With $\mat{T}_2$ being a $d\times p(p + 1) / 2$ full rank matrix and the duplication matrix $\mat{D}_p$ is full rank of dimension $p(p + 1) / 2 \times p^2$ we have $\mat{T}_2\pinv{\mat{D}_p}$ to be $d\times p^2$ of full rank. This means that $\mat{P} = \pinv{(\mat{T}_2\pinv{\mat{D}_p})}\mat{T}_2\pinv{\mat{D}_p}$ is a $p^2\times p^2$ projection of rank $d$ and $\mat{I}_{p^2} - \mat{P}$ is then a projection of rank $p^2 - d$. This leads to
    \begin{displaymath}
        \manifold{CK}_{\mat{\Omega}}
            = \{ \mat{\Omega}\in\manifold{K}_{\mat{\Omega}} : (\mat{I}_{p^2} - \mat{P})\vec{\mat{\Omega}} = \mat{0} \}.
    \end{displaymath}

    To show that $\manifold{CK}_{\mat{\Omega}}$ is an embedded submanifold of $\manifold{K}_{\mat{\Omega}}$ we apply the ``Constant-Rank Level Set Theorem'' \textcite[Thm~5.12]{introToSmoothMani-Lee2012} which states (slightly adapted) the following;
    Let $\manifold{A}$, $\manifold{B}$ be smooth manifolds and $F:\manifold{A}\to\manifold{B}$ a smooth map such that $\nabla_{\mat{a}} F$ has constant matrix rank for all $\mat{a}\in\manifold{A}$. Then, for every $\mat{b}\in F(\mat{A})\subseteq\manifold{B}$, the preimage $F^{-1}(\{ \mat{b} \})$ is a smooth embedded submanifold of $\manifold{A}$.

    In our setting, we have $F:\manifold{K}_{\mat{\Omega}}\to\mathbb{R}^{p^2}$ defined as $F(\mat{\Omega}) = (\mat{I}_{p^2} - \mat{P})\vec{\mat{\Omega}}$ with gradient $\nabla_{\mat{\Omega}} F = \mat{I}_{p^2} - \mat{P}$ of constant rank. Therefore, $F^{-1}(\{\mat{0}\}) = \manifold{CK}_{\mat{\Omega}}$ is an embedded submanifold of $\manifold{K}_{\mat{\Omega}}$.

    Finally, the finite product manifold of embedded submanifolds is embedded in the finite product space of their ambient spaces, that is $\Theta = \mathbb{R}^p \times \manifold{K}_{\mat{B}}\times\manifold{CK}_{\mat{\Omega}} \subset \mathbb{R}^p\times\mathbb{R}^{p\times q}\times\mathbb{R}^{p\times p}$ is embedded.
\end{proof}

\begin{proof}[Proof of \cref{thm:exists-strong-M-estimator-on-subsets}]
    Let $\hat{\mat{\xi}}_n$ be a (weak/strong) M-estimator for the unconstrained problem. This gives by definition, in any case, that
    \begin{displaymath}
        \sup_{\mat{\xi}\in\Xi} M_n(\mat{\xi}) \leq M_n(\hat{\mat{\xi}}_n) + o_P(1).
    \end{displaymath}
    Cause $\emptyset\neq\Theta\subseteq\Xi$ we have $\sup_{\mat{\theta}\in\Theta} M_n(\mat{\theta}) \leq \sup_{\mat{\xi}\in\Xi} M_n(\mat{\xi})$ and with $M_n(\mat{\xi}) < \infty$ for any $\mat{\xi}\in\Xi$
    \begin{displaymath}
        P\Bigl( \sup_{\mat{\theta}\in\Theta} M_n(\mat{\theta}) < \infty \Bigr)
        \geq
        P\Bigl( \sup_{\mat{\xi}\in\Xi} M_n(\mat{\xi}) < \infty \Bigr)
        \xrightarrow{n\to\infty}
        1.
    \end{displaymath}
    If $\sup_{\mat{\theta}\in\Theta} M_n(\mat{\theta}) < \infty$, then, for any $0 < \epsilon_n$ exists $\hat{\mat{\theta}}_n\in\Theta$ such that $\sup_{\mat{\theta}\in\Theta} M_n(\mat{\theta}) - \epsilon_n \leq M_n(\hat{\mat{\theta}}_n)$. Therefore, we can choose $\epsilon_n\in o(n^{-1})$, which yields
    \begin{displaymath}
        P\Bigl( M_n(\hat{\mat{\theta}}_n) \geq \sup_{\mat{\theta}\in\Theta} M_n(\mat{\theta}) - o(n^{-1}) \Bigr)
        \geq
        P\Bigl( \sup_{\mat{\theta}\in\Theta} M_n(\mat{\theta}) < \infty \Bigr)
        \xrightarrow{n\to\infty}
        1.
    \end{displaymath}
    The last statement states
    \begin{displaymath}
        M_n(\hat{\mat{\theta}}_n) \geq \sup_{\mat{\theta}\in\Theta} M_n(\mat{\theta}) - o_P(n^{-1})
    \end{displaymath}
    which is the definition of $\hat{\mat{\theta}}_n$ being a strong M-estimator over $\Theta$.
\end{proof}

\begin{proof}[Proof of \cref{thm:M-estimator-consistency-on-subsets}]
    It follows the proof of \textcite[Proposition~2.4]{asymptoticMLE-BuraEtAl2018} with the same assumptions. The only exception is we only require $\Theta$ to be a subset of $\Xi$. This is accounted for  by replacing Lemma~2.3 in \textcite{asymptoticMLE-BuraEtAl2018} with \cref{thm:exists-strong-M-estimator-on-subsets} to obtain the existence of a strong M-estimator on $\Theta$.
\end{proof}


\begin{proof}[Proof of \cref{thm:M-estimator-asym-normal-on-manifolds}]
    Let $\varphi:U\to\varphi(U)$ be a coordinate chart\footnote{By \cref{def:manifold}, the chart $\varphi : U\to\varphi(U)$ is bi-continuous, is infinitely often continuously differentiable, and has a continuously differentiable inverse $\varphi^{-1} : \varphi(U) \to U$. Furthermore, the domain $U$ is open according to the trace topology on $\Theta$, that means that their exists an open set $O\subseteq\mathbb{R}^p$ such that $U = \Theta\cap O$.} with $\mat{\theta}_0\in U\subseteq\Theta$. As $\varphi$ is continuous we get with the continuous mapping theorem on metric spaces \textcite[Thm~18.11]{asymStats-van_der_Vaart1998} that $\varphi(\hat{\mat{\theta}}_n)\xrightarrow{p}\varphi(\mat{\theta}_0)$ which implies $P(\varphi(\hat{\mat{\theta}}_n)\in\varphi(U))\xrightarrow{n\to\infty}1$.

    The next step is to apply \textcite[Thm~5.23]{asymStats-van_der_Vaart1998} to $\hat{\mat{s}}_n = \varphi(\hat{\mat{\theta}}_n)$. Therefore, assume that $\hat{\mat{s}}_n\in\varphi(U)$. Denote with $\mat{s} = \varphi(\mat{\theta})\in\varphi(U)\subseteq\mathbb{R}^d$ the coordinates of the parameter $\mat{\theta}\in U\subseteq\Theta$ of the $d = \dim(\Theta)$ dimensional manifold $\Theta\subseteq\mathbb{R}^p$. With $\varphi : U\to\varphi(U)$ being bijective, we can express $m_{\mat{\theta}}$ in terms of $\mat{s} = \varphi(\mat{\theta})$ for every $\mat{\theta}\in U$ as $m_{\mat{\theta}} = m_{\varphi^{-1}(\mat{s})}$. Furthermore, denote
    \begin{displaymath}
        M(\mat{\theta}) = \E[m_{\mat{\theta}}(Z)] \qquad\text{and}\qquad M_{\varphi}(\mat{s}) = \E[m_{\varphi^{-1}(\mat{s})}(Z)] = M(\varphi^{-1}(\mat{s})).
    \end{displaymath}

    \begin{figure}[hpt!]
        \centering
        \includegraphics{images/embeddImage.pdf}
        \caption{\label{fig:proof:M-estimator-asym-normal-on-manifolds}Depiction ot the notation used in the proof of \cref{thm:M-estimator-asym-normal-on-manifolds}. Example with $p = 3$ and $d = \dim(\Theta) = 2$.}
    \end{figure}

    By assumption, the function $M(\mat{\theta})$ is twice continuously differentiable in an neighborhood\footnote{A set $N$ is called a neighborhood of $u$ if there exists an open set $O$ such that $u\in O\subseteq N$.} of $\mat{\theta}_0$. W.l.o.g. we can assume that $U$ is contained in that neighborhood. Then, using the chain rule, we get the gradient of $M_{\varphi}$ at $\mat{s}_0$ to be $\mat{0}$ by
    \begin{displaymath}
        \nabla M_{\varphi}(\mat{s}_0) = {\nabla\varphi^{-1}(\mat{s}_0)}{\nabla M(\varphi^{-1}(\mat{s}_0))} = {\nabla\varphi^{-1}(\mat{s}_0)}{\nabla M(\mat{\theta}_0)} = {\nabla\varphi^{-1}(\mat{s}_0)}\mat{0} = \mat{0}
    \end{displaymath}
    because $\mat{\theta}_0 = \varphi^{-1}(\mat{s}_0)$ is a maximizer of $M$. For the second-derivative, evaluated at $\mat{s}_0 = \varphi(\mat{\theta}_0)$, we have
    \begin{displaymath}
        \nabla^2 M_{\varphi}(\mat{s}_0)
            = \nabla\varphi^{-1}(\mat{s}_0)\nabla^2 M(\varphi^{-1}(\mat{s}_0))\t{\nabla\varphi^{-1}(\mat{s}_0)}
            = \nabla\varphi^{-1}(\mat{s}_0)\mat{H}_{\mat{\theta}_0}\t{\nabla\varphi^{-1}(\mat{s}_0)}
    \end{displaymath}
    using $\nabla M_{\varphi}(\mat{s}_0) = \mat{0}$. This gives the second-order Taylor expansion of $M_{\varphi}$ at $\mat{s}_0$ as
    \begin{displaymath}
        M_{\varphi}(\mat{s}) = M_{\varphi}(\mat{s}_0) + \frac{1}{2}\t{(\mat{s} - \mat{s}_0)} \nabla^2 M_{\varphi}(\mat{s}_0) (\mat{s} - \mat{s}_0) + \mathcal{O}(\|\mat{s} - \mat{s}_0\|^3)
    \end{displaymath}
    We also need to check the local Lipschitz condition of $m_{\varphi^{-1}(\mat{s})}$. Therefore, let $V_{\epsilon}(\mat{s}_0) = \{ \mat{s}\in\mathbb{R}^d : \|\mat{s} - \mat{s}_0\| < \epsilon \}$ be the open $\epsilon$-ball with center $\mat{s}_0$. Since $\varphi(U)$ contains $\mat{s}_0$, and is open in $\mathbb{R}^d$, there exists an $\epsilon > 0$ such that $V_{\epsilon}(\mat{s}_0)\subseteq\varphi(U)$. Then, the closed $\epsilon/2$ ball $\overline{V}_{\epsilon / 2}(\mat{s}_0)$ is a neighborhood of $\mat{s}_0$ and the supremum $\sup_{\mat{s}\in \overline{V}_{\epsilon / 2}(\mat{s}_0)}\|\nabla\varphi^{-1}(\mat{s})\| < \infty$ due to the continuouty of $\nabla\varphi^{-1}$ on $\varphi(U)$ with $\overline{V}_{\epsilon / 2}(\mat{s}_0)\subset V_{\epsilon}(\mat{s}_0)\subseteq\varphi(U)$. Then, for almost every $z$ and every $\mat{s}_1 = \varphi(\mat{\theta}_1), \mat{s}_2 = \varphi(\mat{\theta}_2)\in\overline{V}_{\epsilon / 2}(\mat{s}_0)$ holds
    \begin{multline*}
        | m_{\varphi^{-1}(\mat{s}_1)}(z) - m_{\varphi^{-1}(\mat{s}_2)}(z) |
        = | m_{\mat{\theta}_1}(z) - m_{\mat{\theta}_2}(z) |
        \overset{(a)}{\leq} u(z) \| \mat{\theta}_1 - \mat{\theta}_2 \|
        = u(z) \| \varphi^{-1}(\mat{s}_1) - \varphi^{-1}(\mat{s}_2) \| \\
        \overset{(b)}{\leq} u(z) \sup_{\mat{s}\in \overline{V}_{\epsilon / 2}(\mat{s}_0)}\|\nabla\varphi^{-1}(\mat{s})\| \|\mat{s}_1 - \mat{s}_2\|
        =: v(z) \|\mat{s}_1 - \mat{s}_2\|.
    \end{multline*}
    Here, $(a)$ holds by assumption and $(b)$ is a result of the mean value theorem. Now, $v(z)$ is measurable and square integrable as a scaled version of $u(z)$. Finally, with $\varphi$ being one-to-one, we get that $\hat{\mat{s}}_n = \varphi(\hat{\mat{\theta}}_n)$ is a strong M-estimator for $\mat{s}_0 = \varphi(\mat{\theta}_0)$ of the objective $M_{\varphi}$. Now, we apply \textcite[Thm~5.23]{asymStats-van_der_Vaart1998} to get the asymptotic normality of $\hat{\mat{s}}_n$ as
    \begin{displaymath}
        \sqrt{n}(\hat{\mat{s}}_n - \mat{s}_0) \xrightarrow{d} \mathcal{N}_{d}(0, \mat{\Sigma}_{\mat{s}_0})
    \end{displaymath}
    where the $d\times d$ variance-covariance matrix $\mat{\Sigma}_{\mat{s}_0}$ is given by
    \begin{align*}
        \mat{\Sigma}_{\mat{s}_0} &= (\nabla^2 M_{\varphi}(\mat{s}_0))^{-1}\E[\nabla_{\mat{s}} m_{\varphi^{-1}(\mat{s}_0)}(Z)\t{(\nabla_{\mat{s}} m_{\varphi^{-1}(\mat{s}_0)}(Z))}](\nabla^2 M_{\varphi}(\mat{s}_0))^{-1}.
    \end{align*}

    {
        \def\PP{\mat{\varPhi}_{\mat{\theta}_0}}
        \def\EE#1#2{\E[\nabla_{#2} m_{#1}(Z)\t{(\nabla_{#2} m_{#1}(Z))}]}

        An application of the delta method yields
        \begin{displaymath}
            \sqrt{n}(\hat{\mat{\theta}}_n - \mat{\theta}_0)
                = \sqrt{n}(\varphi^{-1}(\hat{\mat{s}}_n) - \varphi^{-1}(\mat{s}_0))
                \xrightarrow{d} \mathcal{N}_p(0, \t{\nabla\varphi^{-1}(\mat{s}_0)}\mat{\Sigma}_{\mat{s}_0}{\nabla\varphi^{-1}(\mat{s}_0)}).
        \end{displaymath}
        We continue by reexpressing the $p\times p$ asymtotic variance-covariance matrix of $\hat{\mat{\theta}}_n$ in terms of $\mat{\theta}_0$ instead of $\mat{s}_0 = \varphi(\mat{\theta}_0)$. Therefore, let $\PP = \t{\nabla\varphi^{-1}(\varphi(\mat{\theta}_0))} = \t{\nabla\varphi^{-1}(\mat{s}_0)}$ and observe that for all $\mat{s}\in\varphi(U)$, the gradient of $\mat{s}\mapsto m_{\varphi^{-1}(\mat{s})}(z)$ evaluated at $\mat{s}_0 = \varphi(\mat{\theta}_0)$ has the form
        \begin{displaymath}
            \nabla_{\mat{s}}m_{\varphi^{-1}(\mat{s}_0)}(z)
            = \nabla\varphi^{-1}(\mat{s}_0)\nabla_{\mat{\theta}}m_{\mat{\theta}_0}(z)
            = \t{\PP}\nabla_{\mat{\theta}}m_{\mat{\theta}_0}(z).
        \end{displaymath}
        Then
        \begin{multline*}
            \t{\nabla\varphi^{-1}(\mat{s}_0)}\mat{\Sigma}_{\mat{s}_0}{\nabla\varphi^{-1}(\mat{s}_0)} \\
            = \t{\nabla\varphi^{-1}(\mat{s}_0)}(\nabla^2 M_{\varphi}(\mat{s}_0))^{-1}\EE{\varphi^{-1}(\mat{s}_0)}{}(\nabla^2 M_{\varphi}(\mat{s}_0))^{-1}{\nabla\varphi^{-1}(\mat{s}_0)} \\
            = {\PP}(\t{\PP}\mat{H}_{\mat{\theta}_0}\PP)^{-1}\t{\PP}\EE{\mat{\theta}_0}{\mat{\theta}}{\PP}(\t{\PP}\mat{H}_{\mat{\theta}_0}\PP)^{-1}\t{\PP} \\
            = \mat{\Pi}_{\mat{\theta}_0}\EE{\mat{\theta}_0}{\mat{\theta}}\mat{\Pi}_{\mat{\theta}_0}
        \end{multline*}
        where the last equality holds by $\Span\PP = T_{\mat{\theta}_0}\Theta$ by \cref{def:tangent-space} of the tangent space $T_{\mat{\theta}_0}\Theta$.

        It remains to show that $\mat{\Pi}_{\mat{\theta}_0} = \mat{P}_{\mat{\theta}_0}\pinv{(\t{\mat{P}_{\mat{\theta}_0}}\mat{H}_{\mat{\theta}_0}\mat{P}_{\mat{\theta}_0})}\t{\mat{P}_{\mat{\theta}_0}}$ for any $p\times k$ matrix $\mat{P}_{\mat{\theta}_0}$ such that $k\geq d$ and $\Span{\mat{P}_{\mat{\theta}_0}} = T_{\mat{\theta}_0}\Theta$. This also ensures that the final result is independent of the chosen chart $\varphi$, since the tangent space does not depend on a specific chart. Therefore, let $\PP = {\mat{Q}}{\mat{R}}$ and $\mat{P}_{\mat{\theta}_0} = \widetilde{\mat{Q}}\widetilde{\mat{R}}$ be their thin QR decompositions, respectively. Both $\mat{Q}, \widetilde{\mat{Q}}$ have dimension $p\times d$ With $\mat{Q}$ being semi-orthogonal, $\mat{R}$ is invertible of dimension $d\times d$ while $\widetilde{\mat{R}}$ is a $d\times k$ full row-rank matrix. With $\mat{Q}$ being semi-orthogonal the $p\times p$ matrix $\mat{Q}\t{\mat{Q}}$ is an orthogonal projection onto $\Span\mat{Q} = \Span\mat{P}_{\mat{\theta}_0} = T_{\mat{\theta}_0}\Theta$. This allows to express $\mat{P}_{\mat{\theta}_0}$ in terms of $\mat{Q}$ as
        \begin{displaymath}
            \mat{P}_{\mat{\theta}_0} = \mat{Q}\t{\mat{Q}}\mat{P}_{\mat{\theta}_0}
                = \mat{Q}\t{\mat{Q}}\widetilde{\mat{Q}}\widetilde{\mat{R}} =: {\mat{Q}}\mat{M}.
        \end{displaymath}
        From $\Span\mat{Q} = \Span\mat{P}_{\mat{\theta}_0}$ follows that the $d\times k$ matrix $\mat{M}$ is also of full row-rank. We get $\mat{M}\pinv{\mat{M}} = \mat{I}_d = \mat{R}\mat{R}^{-1}$ as a property of the Moore-Penrose pseudo inverse with $\mat{M}$ being of full row-rank. Another property of the pseudo inverse is that for matrices $\mat{A}, \mat{B}$, where $\mat{A}$ has full column-rank and $\mat{B}$ has full row-rank, holds $\pinv{(\mat{A}\mat{B})} = \pinv{\mat{B}}\pinv{\mat{A}}$. This enables the computation
        \begin{multline*}
            \mat{P}_{\mat{\theta}_0}\pinv{(\t{\mat{P}_{\mat{\theta}_0}}\mat{H}_{\mat{\theta}_0}\mat{P}_{\mat{\theta}_0})}\t{\mat{P}_{\mat{\theta}_0}}
            = \mat{Q} \mat{M} \pinv{\mat{M}} (\t{\mat{Q}} \mat{H}_{\mat{\theta}_0} \mat{Q})^{-1} \t{(\mat{M} \pinv{\mat{M}})} \t{\mat{Q}} \\
            = \mat{Q} {\mat{R}} {\mat{R}}^{-1} (\t{\mat{Q}} \mat{H}_{\mat{\theta}_0} \mat{Q})^{-1} \t{({\mat{R}} {\mat{R}}^{-1})} \t{\mat{Q}}
            = \PP(\t{\PP}\mat{H}_{\mat{\theta}_0}\PP)^{-1}\t{\PP}
            = \mat{\Pi}_{\mat{\theta}_0}.
        \end{multline*}

    }
\end{proof}



In the following we rewrite the log-likelihood \eqref{eq:log-likelihood} in a simpler form. This simplifies the proof of \cref{thm:asymptotic-normality-gmlm} as well as provides the notation to express the regularity conditions of \cref{thm:asymptotic-normality-gmlm} in a compact form.

Rewriting the first natural parameter component $\mat{\eta}_{1y}$ defined in \eqref{eq:eta1-manifold} gives
\begin{displaymath}
    \mat{\eta}_{1y}
        = \vec{\overline{\ten{\eta}}} + \mat{B}\vec{\ten{F}_y}
        = \mat{I}_p\vec{\overline{\ten{\eta}}} + (\t{(\vec{\ten{F}_y})}\otimes\mat{I}_p)\vec{\mat{B}}
        = \begin{pmatrix}
            \mat{I}_p & \t{(\vec{\ten{F}_y})}\otimes\mat{I}_p
        \end{pmatrix}\begin{pmatrix}
            \vec{\overline{\ten{\eta}}} \\
            \vec{\mat{B}}
        \end{pmatrix}.
\end{displaymath}
For the second natural parameter component $\mat{\eta}_2$, modeled in \eqref{eq:eta2-manifold}, we have
\begin{displaymath}
    \langle \mat{\eta}_2, \mat{T}_2\vech((\vec{\ten{X}})\t{(\vec{\ten{X}})}) \rangle
        = \langle \t{(\mat{T}_2\pinv{\mat{D}_p})}\mat{\eta}_2, \vec(\ten{X}\circ\ten{X}) \rangle
        = \langle c\,\mat{\Omega}, \ten{X}\circ\ten{X} \rangle
\end{displaymath}
which means that
\begin{displaymath}
    c \vec{\mat{\Omega}} = \t{(\mat{T}_2\pinv{\mat{D}_p})}\mat{\eta}_2.
\end{displaymath}
The inverse relation is
\begin{displaymath}
    \mat{\eta}_2 = c\t{(\pinv{(\mat{T}_2\pinv{\mat{D}_p})})}\vec\mat{\Omega} = c\t{(\pinv{(\mat{T}_2\pinv{\mat{D}_p})})}\mat{D}_p\vech\mat{\Omega},
\end{displaymath}
describing the linear relation between $\mat{\eta}_2$ and $\vech{\mat{\Omega}}$. This gives the following relation between $\mat{\eta}_y = (\mat{\eta}_{1y}, \mat{\eta}_2)$ and $\mat{\xi} = (\vec{\overline{\ten{\eta}}}, \vec{\mat{B}}, \vech{\mat{\Omega}})\in\Xi$ as
\begin{equation}
    \mat{\eta}_y = \begin{pmatrix}
        \mat{I}_p & \t{(\vec{\ten{F}_y})}\otimes\mat{I}_p & 0 \\
        0 & 0 & c\t{(\pinv{(\mat{T}_2\pinv{\mat{D}_p})})}\mat{D}_p
    \end{pmatrix}\begin{pmatrix}
        \vec{\overline{\ten{\eta}}} \\
        \vec{\mat{B}} \\
        \vech{\mat{\Omega}}
    \end{pmatrix} =: \mat{F}(y)\mat{\xi} \label{eq:eta-to-xi-linear-relation}
\end{equation}
where $\mat{F}(y)$ is a $(p + d)\times p (p + 2 q + 3) / 2$ dimensional matrix valued function in $y$. Moreover, for every $y$ the matrix $\mat{F}(y)$ is of full rank $p + d$.

The log-likelihood of model \eqref{eq:quad-density} for the unconstrained parameters $\xi\in\Xi$ is
\begin{displaymath}
    l_n(\mat{\xi})
        = \frac{1}{n}\sum_{i = 1}^{n} (\langle \mat{t}(\ten{X}), \mat{\eta}_{y} \rangle - b(\mat{\eta}_y))
        =: \frac{1}{n}\sum_{i = 1}^{n} m_{\mat{\xi}}(Z_i)
\end{displaymath}
where $Z_i = (\ten{X}_i, Y_i)$. Using \eqref{eq:eta-to-xi-linear-relation} we can write
\begin{displaymath}
    m_{\mat{\xi}}(z) = \langle\mat{t}(\ten{X}), \mat{F}(y)\mat{\xi}\rangle - b(\mat{F}(y)\mat{\xi}).
\end{displaymath}

The following are the regularity conditions for the log-likelihood required by \cref{thm:asymptotic-normality-gmlm}.

\begin{condition}\label{cond:differentiable-and-convex}
    The mapping $\mat{\xi}\mapsto m_{\mat{\xi}}(z)$ is twice continuously differentiable for almost every $z$ and $z\mapsto m_{\mat{\xi}}(z)$ is measurable. Moreover, $\mat{\eta}\mapsto b(\mat{\eta})$ is strictly convex. \todo{Furthermore, for every $\widetilde{\mat{\eta}}$ holds $P(\mat{F}(Y)\mat{\xi} = \widetilde{\mat{\eta}}) < 1$. Do I need this???}
\end{condition}

\begin{condition}\label{cond:moments}
    It holds $\E\|\t{\mat{t}(\ten{X})}\mat{F}(Y)\| < \infty$ and $\E\|\t{\mat{t}(\ten{X})}\mat{F}(Y)\|^2 < \infty$.
\end{condition}

\begin{condition}\label{cond:finite-sup-on-compacta}
    The mapping $\mat{\eta}\mapsto b(\mat{\eta})$ is twice continuously differentiable and for every non-empty compact $K\subseteq\Xi$ holds
    \begin{gather*}
        \E\sup_{\mat{\xi}\in K}\|b(\mat{F}(Y)\mat{\xi})\| < \infty, \qquad
        \E\sup_{\mat{\xi}\in K}\|\t{\nabla b(\mat{F}(Y)\mat{\xi})}\mat{F}(Y)\|^2 < \infty, \\
        \E\sup_{\mat{\xi}\in K}\| \t{\mat{F}(Y)}\nabla^2 b(\mat{F}(Y)\mat{\xi})\mat{F}(Y) \| < \infty.
    \end{gather*}
\end{condition}

The following is a technical Lemma used in the proof of \cref{thm:asymptotic-normality-gmlm}.

\begin{lemma}\label{thm:kron-manifold-tangent-space}
    Let $\manifold{A}_k\subseteq\mathbb{R}^{p_k\times q_k}\backslash\{\mat{0}\}$ for $k = 1, \ldots, r$ be smooth embedded submanifolds as well as ether a sphere or a cone. Then
    \begin{displaymath}
        \manifold{K} = \Bigl\{ \bigkron_{k = r}^{1}\mat{A}_k : \mat{A}_k\in\manifold{A}_k \Bigr\}
    \end{displaymath}
    is an embedded manifold in $\mathbb{R}^{p\times q}$ for $p = \prod_{k = 1}^{r} p_k$ and $q = \prod_{k = 1}^{r} q_k$.
    Furthermore, define for $j = 1, \ldots, r$ the matrices
    \begin{equation}\label{eq:kron-differential-span}
        \mat{\Gamma}_j
            = \bigkron_{k = r}^{1}(\mat{I}_{p_k q_k}\mathrm{\ if\ } j = k \mathrm{\ else\ }\vec{\mat{A}_k})
            = \bigkron_{k = r}^{j + 1}(\vec{\mat{A}_k})\otimes\mat{I}_{p_j q_j}\otimes\bigkron_{k = j - 1}^{1}(\vec{\mat{A}_k})
    \end{equation}
    and let $\gamma_j$ be $p_j q_j\times d_j$ matrices with $d_j \geq\dim\manifold{A}_j$ which span the tangent space $T_{\mat{A}_j}\manifold{A}_j$ of $\manifold{A}$ at $\mat{A}_j\in\manifold{A}_j$, that is $\Span\gamma_j = T_{\mat{A}_j}\manifold{A}_j$.
    Then, with the permutation matrix $\mat{S}_{\mat{p}, \mat{q}}$ defined in \eqref{eq:S_pq}, the $p q \times \sum_{k = 1}^{r} d_j$ dimensional matrix
    % Then, with $\mat{p} = (p_1, \ldots, p_r)$, $\mat{q} = (q_1, \ldots, q_r)$ and $\mat{S}_{\mat{p}, \mat{q}}$ being the permutation matrix from \cref{thm:kron-perm}, the $p q \times \sum_{k = 1}^{r} d_j$ dimensional matrix
    \begin{displaymath}
        \mat{P}_{\mat{A}} = \mat{S}_{\mat{p}, \mat{q}}\left[\mat{\Gamma}_1\mat{\gamma}_1, \mat{\Gamma}_2\mat{\gamma}_2, \ldots, \mat{\Gamma}_r\mat{\gamma}_r\right]
    \end{displaymath}
    spans the tangent space $T_{\mat{A}}\manifold{K}$ of $\manifold{K}$ at $\mat{A} = \bigkron_{k = r}^{1}\mat{A}_k\in\manifold{K}$, in formula $\Span\mat{P}_{\mat{A}} = T_{\mat{A}}\manifold{K}$.
\end{lemma}
\begin{proof}
    % The statement that $\manifold{K}$ is an embedded manifold as well as its dimension follows via induction using \cref{thm:kron-manifolds}.
    The statement that $\manifold{K}$ is an embedded manifold follows via induction using \cref{thm:kron-manifolds}.

    We compute the differential of the vectorized Kronecker product using \cref{thm:kron-perm} where $\mat{S}_{\mat{p}, \mat{q}}$ is the permutation \eqref{eq:S_pq} defined therein.
    \begin{multline*}
        \d\vec\bigotimes_{k = r}^{1}\mat{A}_k
             = \vec\sum_{j = 1}^{r}\bigkron_{k = r}^{1}(\ternary{k = j}{\d\mat{A}_j}{\mat{A}_k}) \\
             = \mat{S}_{\mat{p}, \mat{q}}\vec\sum_{j = 1}^{r}\Bigl(\bigouter_{k = 1}^{r}(\ternary{k = j}{\d\mat{A}_j}{\mat{A}_k})\Bigr)
             = \mat{S}_{\mat{p}, \mat{q}}\sum_{j = 1}^{r}\bigkron_{k = r}^{1}(\ternary{k = j}{\vec\d\mat{A}_j}{\vec\mat{A}_k}) \\
             = \mat{S}_{\mat{p}, \mat{q}}\sum_{j = 1}^{r}\Bigl(\bigkron_{k = r}^{1}(\ternary{k = j}{\mat{I}_{p_j q_j}}{\vec\mat{A}_k})\Bigr)\vec\d\mat{A}_j
             = \mat{S}_{\mat{p}, \mat{q}}\sum_{j = 1}^{r}\mat{\Gamma}_j\vec\d\mat{A}_j \\
             = \mat{S}_{\mat{p}, \mat{q}}[\mat{\Gamma}_1, \ldots, \mat{\Gamma}_r]\begin{pmatrix}
                \vec\d\mat{A}_1 \\ \vdots \\ \vec\d\mat{A}_r
            \end{pmatrix}
    \end{multline*}
    Due to the definition of the manifold this differential provides the gradient of a surjective map into the manifold. The span of the gradient then spans the tangent space.

    Now, we take a closer look at the differentials $\vec{\d\mat{A}_j}$ for $j = 1, \ldots, r$. Let $\varphi_j$ be a chart of $\manifold{A}_j$ in a neighborhood of $\mat{A}_j$. Then, $\mat{A}_j = \varphi_j^{-1}(\varphi_j(\mat{A}_j))$ which gives
    \begin{displaymath}
        \vec{\d\mat{A}_j} = \t{\nabla\varphi_j^{-1}(\varphi_j(\mat{A}_j))}\vec\d\varphi_j(\mat{A}_j).
    \end{displaymath}
    Therefore, for every matrix $\mat{\gamma}_j$ such that $\Span{\mat{\gamma}_j} = T_{\mat{A}_j}\manifold{A}_j$ holds $\Span{\t{\nabla\varphi_j^{-1}(\varphi_j(\mat{A}_j))}} = \Span{\mat{\gamma}_j}$ by \cref{def:tangent-space} of the tangent space. We get
    \begin{displaymath}
        \Span\mat{S}_{\mat{p}, \mat{q}}[\mat{\Gamma}_1, \ldots, \mat{\Gamma}_r]\begin{pmatrix}
            \vec\d\mat{A}_1 \\ \vdots \\ \vec\d\mat{A}_r
        \end{pmatrix}
        =
        \Span\mat{S}_{\mat{p}, \mat{q}}[\mat{\Gamma}_1\mat{\gamma}_1, \ldots, \mat{\Gamma}_r\mat{\gamma}_r]
        =
        \Span\mat{P}_{\mat{A}}
    \end{displaymath}
    which concludes the proof.
\end{proof}


\begin{proof}[Proof of \cref{thm:asymptotic-normality-gmlm}]
    The proof consists of three parts. First, we show the existence of a consistent strong M-estimator by applying \cref{thm:M-estimator-consistency-on-subsets}. Next, we apply \cref{thm:M-estimator-asym-normal-on-manifolds} to obtain its asymptotic normality. We conclude by computing the missing parts of the asymtotic covariance matrix $\mat{\Sigma}_{\mat{\theta}_0}$ provided by \cref{thm:M-estimator-asym-normal-on-manifolds}.

    We check whether the conditions of \cref{thm:M-estimator-consistency-on-subsets} are satisfied. On $\Xi$, the mapping $\mat{\xi}\mapsto m_{\mat{\xi}}(z) = m_{\mat{\xi}}(\ten{X},y) = \langle \mat{F}(y)\mat{\xi}, \mat{t}(\ten{X}) \rangle - b(\mat{F}(y)\mat{\xi})$ is strictly concave for every $z$ because $\mat{\xi}\mapsto\mat{F}(y)\mat{\xi}$ is linear and $b$ is strictly convex by \cref{cond:differentiable-and-convex}. Since $\ten{X} \mid Y$ is distributed according to \eqref{eq:quadratic-exp-fam}, the function $M(\mat{\xi}) = \E m_{\mat{\xi}}(Z)$ is well defined by \cref{cond:moments}. Let $\mat{\xi}_k = (\vec{\overline{\ten{\eta}}_k}, \vec{\mat{B}_k}, \vech{\mat{\Omega}_k})$, and $f_{\mat{\xi}_k}$ be the pdf of $\ten{X} \mid Y$ indexed by $\mat{\xi}_k$, for $k = 1, 2$. If $\mat{\xi}_1\ne \mat{\xi}_2$, then $f_{\mat{\xi}_1} \neq f_{\mat{\xi}_2}$, which obtains that the true $\mat{\theta}_0$ is a unique maximizer of $\mat{\theta}_0\in\Theta\subseteq\Xi$ by applying \textcite[Lemma~5.35]{asymStats-van_der_Vaart1998}. Finally, under \cref{cond:finite-sup-on-compacta}, all assumptions of \cref{thm:M-estimator-consistency-on-subsets} are fulfilled yielding the existence of an consistent strong M-estimator over $\Theta\subseteq\Xi$.

    Next, let $\hat{\mat{\theta}}_n$ be a strong M-estimator on $\Theta\subseteq\Xi$, whose existence and consistency was shown in the previous step. Since $z\mapsto m_{\mat{\xi}}(z)$ is measurable for all $\mat{\xi}\in\Xi$, it is also measurable in a neighborhood of $\mat{\theta}_0$. The differentiability of  $\mat{\theta}\mapsto m_{\mat{\theta}}(z)$ is stated in \cref{cond:differentiable-and-convex}. For the Lipschitz condition, let $K\subseteq\Xi$ be a compact neighborhood of $\mat{\theta}_0$, which exists since $\Xi$ is open. Then,
    \begin{align*}
        \left| m_{\mat{\theta}_1}(z) - m_{\mat{\theta}_2}(z) \right|
        &= \left| \langle \mat{t}(\ten{X}), \mat{F}(y)(\mat{\theta}_1 - \mat{\theta}_2) \rangle - b(\mat{F}(z)\mat{\theta}_1) + b(\mat{F}(z)\mat{\theta}_2) \right| \\
        &\leq (\| \t{\mat{F}(y)}\mat{t}(\ten{X}) \|_2 + \sup_{\mat{\theta}\in K}\| \nabla b(\mat{F}(y)\mat{\theta}) \mat{F}(y)\| ) \| \mat{\theta}_1 - \mat{\theta}_2 \|_2
        =: u(z)\| \mat{\theta}_1 - \mat{\theta}_2 \|_2
    \end{align*}
    with $u(z)$ being measurable and square integrable derives from \cref{cond:finite-sup-on-compacta}. The existence of a second-order Taylor expansion of $\mat{\theta}\mapsto M(\mat{\theta}) = \E m_{\mat{\theta}}(Z)$ in a neighborhood of $\mat{\theta}_0$ holds by \cref{cond:finite-sup-on-compacta}. Moreover, the Hessian $\mat{H}_{\mat{\theta}_0}$ is non-singular by the strict convexity of $b$ stated in \cref{cond:differentiable-and-convex}. Now, we can apply \cref{thm:M-estimator-asym-normal-on-manifolds} to obtain the asymptotic normality of $\sqrt{n}(\hat{\mat{\theta}}_n - \mat{\theta}_0)$ with variance-covariance structure 
    \begin{equation}\label{eq:asymptotic-covariance-gmlm}
        \mat{\Sigma}_{\mat{\theta}_0} = \mat{\Pi}_{\mat{\theta}_0} \E[\nabla m_{\mat{\theta}_0}(Z)\t{(\nabla m_{\mat{\theta}_0}(Z))}]\mat{\Pi}_{\mat{\theta}_0}
    \end{equation}
    where $\mat{\Pi}_{\mat{\theta}_0} = \mat{P}_{\mat{\theta}_0}(\t{\mat{P}_{\mat{\theta}_0}}\mat{H}_{\mat{\theta}_0}\mat{P}_{\mat{\theta}_0})^{-1}\t{\mat{P}_{\mat{\theta}_0}}$ and $\mat{P}_{\mat{\theta}_0}$ is any $p\times \dim(\Theta)$ matrix such that it spans the tangent space of $\Theta$ at $\mat{\theta}_0$. That is, $\Span \mat{P}_{\mat{\theta}_0} = T_{\mat{\theta}_0}\Theta$.

    Finally, we compute a matrix $\mat{P}_{\mat{\theta}_0}$ such that $\Span{\mat{P}_{\mat{\theta}_0}} = T_{\mat{\theta}_0}\Theta$ for $\Theta = \mathbb{R}^p\times\manifold{K}_{\mat{B}}\times\manifold{CK}_{\mat{\Omega}}$ as in \cref{thm:param-manifold}. Since the manifold $\Theta$ is a product manifold we get a block diagonal structure for $\mat{P}_{\mat{\theta}_0}$ as
    \begin{displaymath}
        \mat{P}_{\mat{\theta}_0} = \begin{pmatrix}
            \mat{I}_p & 0 & 0 \\
            0 & \mat{P}_{\mat{B}_0} & 0 \\
            0 & 0 & \mat{P}_{\mat{\Omega}_0}
        \end{pmatrix}
    \end{displaymath}
    where $\mat{I}_p$ is the identity matrix spanning the tangent space of $\mathbb{R}^p$, which is identified with $\mathbb{R}^p$ itself. The blocks $\mat{P}_{\mat{B}_0}$ and $\mat{P}_{\mat{\Omega}_0}$ need to span the tangent spaces of $\manifold{K}_{\mat{B}}$ and $\manifold{CK}_{\mat{\Omega}}$, respectively. Both $\manifold{K}_{\mat{B}}$ and $\manifold{CK}_{\mat{\Omega}}$ are manifolds according to \cref{thm:kron-manifolds} under the cone condition. The constraint manifold $\manifold{CK}_{\mat{\Omega}}$ is the intersection of $\manifold{K}_{\mat{\Omega}}$ with the span of the projection $\pinv{(\mat{T}_2\pinv{\mat{D}_p})}\mat{T}_2\pinv{\mat{D}_p}$ meaning that the differential $\vec{\d\mat{\Omega}}$ on $\manifold{CK}_{\mat{\Omega}}$ fulfills $\vec{\d\mat{\Omega}} = \pinv{(\mat{T}_2\pinv{\mat{D}_p})}\mat{T}_2\pinv{\mat{D}_p}\vec{\d\mat{\Omega}}$. Now, we can apply \cref{thm:kron-manifold-tangent-space} for $\manifold{K}_{\mat{B}}$ and $\manifold{K}_{\mat{\Omega}}$ which give
    \begin{align*}
        \mat{P}_{\mat{B}_0} &= \mat{S}_{\mat{p}, \mat{q}}[\mat{\Gamma}_{\mat{\beta}_1}\mat{\gamma}_{\mat{\beta}_1}, \ldots, \mat{\Gamma}_{\mat{\beta}_r}\mat{\gamma}_{\mat{\beta}_r}], \\
        \mat{P}_{\mat{\Omega}_0} &= \pinv{(\mat{T}_2\pinv{\mat{D}_p})}\mat{T}_2\pinv{\mat{D}_p}\mat{S}_{\mat{p}, \mat{p}}[\mat{\Gamma}_{\mat{\Omega}_1}\mat{\gamma}_{\mat{\Omega}_1}, \ldots, \mat{\Gamma}_{\mat{\Omega}_r}\mat{\gamma}_{\mat{\Omega}_r}]
    \end{align*}
    where the matrices $\mat{S}_{\mat{p}, \mat{q}}$, $\mat{\Gamma}_{\mat{\beta}_j}$, $\mat{\gamma}_{\mat{\beta}_j}$, $\mat{\Gamma}_{\mat{\Omega}_j}$ and $\mat{\gamma}_{\mat{\Omega}_j}$ are described in \cref{thm:kron-manifold-tangent-space} for the Kronecker manifolds $\manifold{K}_{\mat{B}}$ and $\manifold{K}_{\mat{\Omega}}$. Leading to
    \begin{equation}\label{eq:param-manifold-span}
        \mat{P}_{\mat{\theta}_0} = \begin{pmatrix}
            \mat{I}_p & 0 & 0 \\
            0 & \mat{S}_{\mat{p}, \mat{q}}[\mat{\Gamma}_{\mat{\beta}_1}\mat{\gamma}_{\mat{\beta}_1}, \ldots, \mat{\Gamma}_{\mat{\beta}_r}\mat{\gamma}_{\mat{\beta}_r}] & 0 \\
            0 & 0 & \pinv{(\mat{T}_2\pinv{\mat{D}_p})}\mat{T}_2\pinv{\mat{D}_p}\mat{S}_{\mat{p}, \mat{p}}[\mat{\Gamma}_{\mat{\Omega}_1}\mat{\gamma}_{\mat{\Omega}_1}, \ldots, \mat{\Gamma}_{\mat{\Omega}_r}\mat{\gamma}_{\mat{\Omega}_r}]
        \end{pmatrix}.
    \end{equation}
\end{proof}



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