Vectors are write as boldface lowercase letters (e.g. $\mat a$, $\mat b$), matrices use boldface uppercase or Greek letters (e.g. $\mat A$, $\mat B$, $\mat\alpha$, $\mat\Delta$). The identity matrix of dimensions $p\times p$ is denoted by $\mat{I}_p$ and the commutation matrix as $\mat{K}_{p, q}$ or $\mat{K}_p$ is case of $p = q$. Tensors, meaning multi-dimensional arrays of order at least 3, use uppercase calligraphic letters (e.g. $\ten{A}$, $\ten{B}$, $\ten{X}$, $\ten{Y}$, $\ten{F}$). Boldface indices (e.g. $\mat{i}, \mat{j}, \mat{k}$) denote multi-indices $\mat{i}=(i_1, ..., i_r)\in[\mat{d}]$ where the bracket notation is a shorthand for $[r]=\{1, ..., r\}$ which in conjunction with a multi-index as argument means $[\mat{d}]=[d_1]\times ... \times[d_K]$.
Let $\ten{A}=(a_{i_1,...,i_r})\in\mathbb{R}^{d_1\times ...\times d_r}$ be an order\footnote{Also called 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 axis of $\ten{A}$. For matrices $\mat{B}_k\in\mathbb{R}^{p_k\times d_k}$ with $k\in[r]=\{1, 2, ..., r\}$ the \emph{multi-linear multiplication} is defined element wise as
which results in an order $r$ tensor of dimensions $p_1\times ...\times p_k)$. With this the \emph{$k$-mode product} between the tensor $\ten{A}$ with the matrix $\mat{B}_k$ is given by
Furthermore, the notation $\ten{A}\times_{k\in S}$ is a short hand for writing the iterative application if the mode product for all indices in $S\subset[r]$. 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 products commutes for different modes $j\neq k\Rightarrow\ten{A}\times_j\mat{B}_j\times_k\mat{B}_k =\ten{A}\times_k\mat{B}_k\times_j\mat{B}_j$.
The \emph{inner product} between two tensors of the same order and dimensions is
with which the \emph{Frobenius Norm}$\|\ten{A}\|_F =\sqrt{\langle\ten{A}, \ten{A}\rangle}$. Of interest is also the \emph{maximum norm}$\|\ten{A}\|_{\infty}=\max_{i_1, ..., i_K} a_{i_1, ..., i_K}$. Furthermore, the Frobenius and maximum norm are also used for matrices while for a vector $\mat{a}$ the \emph{2 norm} is $\|\mat{a}\|_2=\sqrt{\langle\mat{a}, \mat{a}\rangle}$.
Matrices and tensor can be \emph{vectorized} by the \emph{vectorization} operator $\vec$. 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 an particular mode. For a tensor $\ten{A}$ of order $r$ and dimensions $d_1, ..., d_r$ the $k$-mode unfolding $\ten{A}_{(k)}$ is a $d_k\times\prod_{l=1, l\neq k}d_l$ matrix. For the tensor $\ten{A}=(a_{i_1,...,i_r})\in\mathbb{R}^{d_1, ..., d_r}$ the elements of the $k$ unfolded tensor $\ten{A}_{(k)}$ are
The rank of a tensor $\ten{A}$ of dimensions $d_1\times ...\times d_r$ is given by a vector $\rank{\ten{A}}=(a_1, ..., a_r)\in[d_1]\times...\times[d_r]$ where $a_k =\rank(\ten{A}_{(k)})$ is the usual matrix rank of the $k$ unfolded tensor.
\begin{theorem}[Tensor Normal to Multi-Variate Normal equivalence]
For a multi-dimensional random variable $\ten{X}$ of order $r$ with dimensions $p_1\times ..., p_r$. Let $\ten{\mu}$ be the mean of the same order and dimensions as $\ten{X}$ and the mode covariance matrices $\mat{\Delta}_i$ of dimensions $p_i\times p_i$ for $i =1, ..., n$. Then the tensor normal distribution is equivalent to the multi-variate normal distribution by the relation
A straight forward way is to rewrite the Tensor Normal density as the density of a Multi-Variate Normal distribution depending on the vectorization of $\ten{X}$. First consider
Next, using a property of the determinant of a Kronecker product $|\mat{\Delta}_1\otimes\mat{\Delta}_2| = |\mat{\Delta}_1|^{p_2}|\mat{\Delta}_2|^{p_1}$ yields
When sampling from the Multi-Array Normal one way is to sample from the Multi-Variate Normal and then reshaping the result, but this is usually very inefficient because it requires to store the multi-variate covariance matrix which is very big. Instead, it is more efficient to sample $\ten{Z}$ as a tensor of the same shape as $\ten{X}$ with standard normal entries and then transform the $\ten{Z}$ to follow the Multi-Array Normal as follows
% Note that the log-likelihood estimate $\hat{l}$ only depends on $\mat\alpha, \mat\beta$. Next, we compute the gradient for $\mat\alpha$ and $\mat\beta$ of $\hat{l}$ used to formulate a Gradient Descent base estimation algorithm for $\mat\alpha, \mat\beta$ as the previous algorithmic. The main reason is to enable an estimation for bigger dimensions of the $\mat\alpha, \mat\beta$ coefficients since the previous algorithm does \emph{not} solve the high run time problem for bigger dimensions.
% Start with the general case of $\mat X_i|(Y_i = y_i)$ is multivariate normal distributed with the covariance $\mat\Delta$ being a $p q\times p q$ positive definite symmetric matrix \emph{without} an further assumptions. We have $i = 1, ..., n$ observations following
% Substitution of the MLE estimates into the log-likelihood $l(\mat\mu, \mat\Delta, \mat\alpha, \mat\beta)$ gives the estimated log-likelihood $\hat{l}(\mat\alpha, \mat\beta)$ as
% We are interested in the gradients $\nabla_{\mat\alpha}\hat{l}(\mat\alpha, \mat\beta)$, $\nabla_{\mat\beta}\hat{l}(\mat\alpha, \mat\beta)$ of the estimated log-likelihood. Therefore, we consider the differential of $\hat{l}$.
% Now, substitution of $\d\mat{r}_i$ into \eqref{eq:deriv1} gives the gradients (not dimension standardized versions of $\D\hat{l}(\mat\alpha)$, $\D\hat{l}(\mat\beta)$) by identification of the derivatives from the differentials (see: \todo{appendix})
% These quantities are very verbose as well as completely unusable for an implementation. By detailed analysis of the gradients we see that the main parts are only element permutations with a high sparsity. By defining the following compact matrix
% \begin{equation}\label{eq:permTransResponse}
% \mat G = \vec^{-1}_{q r}\bigg(\Big( \sum_{j = 1}^n \vec\mat{f}_{y_j}\otimes \widehat{\mat\Delta}^{-1}\mat{r}_j \Big)_{\pi(i)}\bigg)_{i = 1}^{p q k r}{\color{gray}\qquad(q r \times p k)}
% \end{equation}
% with $\pi$ being a permutation of $p q k r$ elements corresponding to permuting the axis of a 4D tensor of dimensions $p\times q\times k\times r$ by $(2, 4, 1, 3)$. As a generalization of transposition this leads to a rearrangement of the elements corresponding to the permuted 4D tensor with dimensions $q\times r\times p\times k$ which is then vectorized and reshaped into a matrix of dimensions $q r \times p k$. With $\mat G$ the gradients simplify to \todo{validate this mathematically}
Let $\ten{X}, \ten{F}$ be order $r$ tensors of dimensions $p_1\times ... \times p_r$ and $q_1\times ... \times q_r$, respectively. We assume the population model
where the error tensor $\epsilon$ is of the same order and dimensions as $\ten{X}$. The distribution of the error tensor is assumed to be mean zero tensor distributed $\ten{\epsilon}\sim\mathcal{TN}(0, \mat{\Delta}_1, ..., \mat{\Delta}_r)$ for symmetric, positive definite covariance matrices $\mat{\Delta}_j$, $j =1, ..., r$.
Given $i =1, ..., n$ i.i.d. observations $(\ten{X}_i, \ten{F}_i)$, the sample model analog is
where the model tensors $\ten{X}, \ten{F}$ collect all observations on an additional sample axis in the last mode, making them tensors of order $r +1$. Meaning that $\ten{X}$, $\ten{\mu}$ and $\ten{\epsilon}$ have dimensions $p_1\times ...\times p_r\times n$ and $\ten{F}$ is of dimensions $q_1\times ...\times q_r\times n$. The mean tensor $\ten{\mu}$ replicates its entries $\ten{\mu}_i =\ten{\mu}_1$, $i =1, ..., n$. Let the estimated residual tensor be
The least squares estimates for $\mat{\alpha}_j$, $j =1, ..., r$ given $n$ i.i.d. observations $(\ten{X}_i, \ten{F}_i)$ are the solution to the minimization problem
With the identities $\|\ten{A}\|_F^2=\tr(\ten{A}_{(j)}\t{\ten{A}_{(j)}})$ and $(\ten{A}\times_j\mat{B})_{(j)}=\mat{B}\ten{A}_{(j)}$ for any $j$ it followds that the differential of the Frobenius norm is equal to
\begin{example}[Vector Valued LS ($r =1$)]\label{ex:ls_vector_case}
Considering the vector valued case ($r =1$), then the sample tensors $\ten{F}=\ten{F}_{(1)}=\t{\mat{F}}$ and $\ten{X}=\ten{X}_{(1)}=\t{\mat{X}}$ which are both matrices of dimensions $n\times p$ and $n\times q$, respectively. The LS estimate for the single parameter matrix $\mat{\alpha}=\mat{\alpha}_1$ is
By the definition of the Tensor Normal, with $p =\prod_{j =1}^r p_j$, we get for $n$ i.i.d. observations $\ten{X}_i, \ten{F}_i$ the log-likelihood in terms of the residuals as
Note that the log-likelihood depends not only on the covariance matrices $\mat{\Delta}_j$, $j =1, ..., r$ but also on the parameter matrices $\mat{\alpha}_j$, $j =1, ..., r$ through the residuals $\ten{R}$ (mean $\mu=0$ is assumed).
Using $\d\log|\mat{A}| =\tr(\mat{A}^{-1}\d\mat{A})$ and $\d\mat{A}^{-1}=-\mat{A}^{-1}(\d\mat{A})\mat{A}^{-1}$ as well as $\langle\ten{A}, \ten{B}\rangle=\tr(\ten{A}_{(j)}\t{\ten{B}_{(j)}})$ for any $j =1, ..., r$ we get the differential of the log-likelihood as
Solving for $\mat{\alpha}_j$ in conjunction with the MLE estimates for the $\mat{\Delta}_j$'s gives a cross dependent system of equations for the $\mat{\alpha}_j$ MLE estimates
Note the similarity to the LS estimates but also that they are \emph{not} identical. Its a well known fact that the LS and MLE estimates under the Multivariate Normal model are identical. This seems to be violated, but this is \emph{not} the case because the equivalency only holds for the unstructured case. Both the LS and MLE solutions simplify in the unstructured case ($\ten{X}_i, \ten{F}_i$ are of order $r =1$, e.g. vector valued) to the same well known solution, compare Example~\ref{ex:ls_vector_case} and Example~\ref{ex:mle_vector_case}.
Like in Example~\ref{ex:ls_vector_case} let the observations be vector valued. We get $\ten{F}=\ten{F}_{(1)}=\t{\mat{F}}$ and $\ten{X}=\ten{X}_{(1)}=\t{\mat{X}}$ which are both matrices of dimensions $n\times p$ and $n\times q$, respectively. The estimated residuals are $\ten{R}=\ten{R}_{(1)}=\t{\mat{R}}=\t{(\mat{X}-\mat{F}\t{\widehat{\mat{\alpha}}})}$ with $\mat{\alpha}=\mat{\alpha}_1$ as the single parameters matrix. In this case the tensor MLE estimate for $\mat{\alpha}$ simplifies to its well known form
\paragraph{Comparison to the general case:}\todo{There are two main differences, first the general case has a closed form solution for the gradient due to the explicit nature of the MLE estimate of $\widehat{\mat\Delta}$ compared to the mutually dependent MLE estimates $\widehat{\mat\Delta}_1$, $\widehat{\mat\Delta}_2$. On the other hand the general case has dramatically bigger dimensions of the covariance matrix ($p q \times p q$) compared to the two Kronecker components with dimensions $q \times q$ and $p \times p$. This means that in the general case there is a huge performance penalty in the dimensions of $\widehat{\mat\Delta}$ while in the other case an extra estimation is required to determine $\widehat{\mat\Delta}_1$, $\widehat{\mat\Delta}_2$.}
A traight forward idea for parameter estimation is to use Gradient Descent. For pure algorithmic speedup, by only changin the update rule but \emph{not} the gradient computation of the objective function, we use Nesterov Accelerated Gradient Descent described in Section~\ref{sec:alg_gradient_descent}. An alternative approach applicable for all the methods is to resolve the cross dependence in the estimator equation systems by assuming all the other estimators to be fixed. This leads to an artificialy created closed form solution for the current estimate which is computed according the closed form solution. By cyclic iterating through all the parameters and iterating this process till convergence we get an alternative method as described in Section~\ref{sec:alg_iterative_updating}. In both cases initial estimates are needed for starting the iterative process which is the subject of Section~\ref{sec:alg_init}.
In Section~\ref{sec:kron_cov} we derived for different objective functions, meaning parameterized functions as minimization target, the gradients. In Section~\ref{sec:ls} the objective function is the Frobenius norm of the estimated residuals for solving the Least Squares problem, then in Section~\ref{sec:mle} its the log-likelihood to find the MLE estimates and in Section~\ref{sec:approx} we had a pseduo log-likelihood. Regardles of which estimates we want to find, denote with $l$ the minimization objective corresponding to the desired minimization problem with parameters $\mat{\Theta}$ collecting all the parameters of the objective. The classic gradient descent algorithm starts with initial values $\mat{\Theta}^{(0)}$, see Section~\ref{sec:alg_init}, and applies gradient updates with a given learning rate $\delta > 0$ untill convergence. The algorithm used is an extention of the classic Gradient Descent algorithm namely Nesterov Accelerated Gradient Descent. This algorithm performs similar to Gradient Descent gradient updates but before evaluation of the gradient an extrapolation of the current position into the previous step direction is performed. Furthermore, an internal line search loop is used to determin an appropriate step size. See Algorithm~\ref{alg:gd} for the case of joint parameter matrices $\widehat{\mat{\alpha}}_1, ..., \widehat{\mat{\alpha}}_r$ and covariances $\widehat{\mat{\Delta}}_1, ..., \widehat{\mat{\Delta}}_r$ estimation. In case that the parameter matrices and the covariances are \emph{not} estimated together, like in the LS estimation, the parameter vector $\mat{\Theta}$ consists only of the parameter matrices $\mat{\alpha}_j$ (which is the only difference) and at the end of the algorithm the estimated parameter matrices can be used for estimation of the covariances.
\todo{more details, better explanation, higher/lower abstraction with respect to the different methods?!}
For both the LS and MLE estimates we have derived equation systems, namely \eqref{eq:ls_est_alphas}, \eqref{eq:mle_est_alphas} and \eqref{eq:mle_est_Deltas}, for the estimates which are cross dependent. The idea of Iterative Cyclic Updating is simply to take the cross dependent equations and assume the unknown quantities given from the previous iteration. Then the cross dependency reduces to a closed form solution depending on the known previous iterates. This iterative prodess is repeated till convergence, see Algorithm~\ref{alg:iterative_updating} for the case of LS.
\begin{algorithm}[ht]
\caption{\label{alg:iterative_updating}Iterative Cyclic Updating for LS estimates}
\begin{algorithmic}[1]
\State Arguments: Order $r +1$ tensors $\ten{X}$, $\ten{F}$
A refined version would be to always take the newest estimates. In the case of Algorithm~\ref{alg:iterative_updating} this means that when computing $\widehat{\mat{\alpha}}_j^{(t +1)}$ we use $\widehat{\mat{\alpha}}_k^{(t +1)}$ for $k =1, ..., j -1$ and $\widehat{\mat{\alpha}}_k^{(t)}$ for $k = j +1, ..., r$ in line \ref{alg:iterative_updating:update} instead.
Furthermore, there is also the idea of randomizing the updating order which seems improve convergence and kind of stabalizes the algorithm.
Currently there are two approaches for the initial value estimates required by Algorithm~\ref{alg:gd} and Algorithm~\ref{alg:iterative_updating}. First there is the \emph{Van Loan and Pitsianis} (VLP) method shortly described in Section~\ref{sec:VLP}. The second ansatz is to perform an \emph{Higher Order Principal Component Analysis} (HOPCA) and take the required amount of eigvectors as initial values described in Section~\ref{sec:HOPCA}.
\subsubsection{Van Loan and Pitsianis (VLP)}\label{sec:VLP}
The VLP approach builds on an least squares solution on an rank 1 Kronecker product decomposition \cite{ApproxKron-VanLoanPitsianis1993}. To use the VLP decomposition we solve the vectorized version of model \ref{eq:sample_model} (assuming $\ten{\mu}=0$)
Let $\mat{B}=\bigotimes_{j = r}^1\t{\mat{\alpha}}_j$, then an estimate of $\mat{B}$ without considering the Kronecker structure is given by the usual least squres solution
Then approximate the $\widehat{\mat{B}}$ using the VLP rank 1 Kronecker decomposition (iteratively if rank $r > 2$, e.g. when there are more than two $\widehat{\mat{\alpha}}_j$'s) as
\subsubsection{Higher Order Principal Component Analysis (HOPCA)}\label{sec:HOPCA}
The \emph{Higher Order Principal Component Analysis} a simple estimation method for estimationg Principal Components for each mode of a dataset consisting of tensor valued observations as illustrated in Algorithm~\ref{alg:HOPCA}.
By using initial values from Section~\ref{sec:alg_init} for initial values for the Gradiet Descent algorithm from Section~\ref{sec:alg_gradient_descent} or the Iterative Updating algorithm from Section~\ref{sec:alg_iterative_updating} the complete sequence of algorithm for estimation is combined as ilustrated in Figure~\ref{fig:algo_dependency}.
\node[draw, text centered, text width = 3cm, gray] (vlp) at (-2, 0) {VLP \\ Section~\ref{sec:alg_init}};
\node[draw, text centered, text width = 3cm] (hopca) at (2, 0) {HOPCA \\ Section~\ref{sec:alg_init}};
\node[draw, text centered, text width = 3cm] (ls) at (0, -1.5) {LS \\ Section~\ref{sec:ls},~\ref{sec:alg_gradient_descent},~\ref{sec:alg_iterative_updating}};
\node[draw, text centered, text width = 3cm] (mle) at (0, -3) {MLE \\ Section~\ref{sec:mle},~\ref{sec:alg_gradient_descent},~\ref{sec:alg_iterative_updating}};
% \node[draw, text centered, text width = 3cm] (approx) at (2, -3) {pseudo MLE \\ Section~\ref{sec:mle},~\ref{sec:alg_gradient_descent},\todo{}};
For measring the performance of the different methods in the simulations we employ different metrics to compare the estimates against the ``true'' parameters used to generate the simulation data. The parameters in the sample model \ref{eq:sample_model} are the parameter matrices $\widehat{\mat{\alpha}}_j$ and the covariance matrices $\widehat{\mat{\Delta}}_j$. Given the ``true'' parameters $\mat{\alpha}_j, \mat{\Delta}_j$, $j =1, ..., r$ used to generate the simulation data samples $(\ten{X}_i, \ten{F}_i)$ for $i =1, ..., n$ we employ a few metrics.
First there is the \emph{maximum subspace distance} as the maximum over the $j =1, ..., r$ modes of the Frobenius norms of the subspace projection matrix differences;
where $\mat{P}_{\mat{A}}=\mat{A}(\t{\mat{A}}\mat{A})^{-1}\t{\mat{A}}$ is the projection onto $\Span(\mat{A})$.
Another interesting distance is between the Kronecker products of the parameter matrices $\mat{\beta}=\bigotimes_{j = r}^1\mat{\alpha}_j$ and its estimate $\widehat{\mat{\beta}}=\bigotimes_{j = r}^1\widehat{\mat{\alpha}}_j$. They correspond to the parameters of the vectorized model under the Kronecker product constraint. We use again the Frobenius norm of the projection differences
This might get very expensive to compute directly. But fortunetly this can be drastically simplified, in the sense of the size of the involved matrices. Therefore, we use some properties of projections as well as the Kronecker product which allow to rewrite
This formulation allows for substantialy more efficient implementation which can lead to a drastic speedup in the simulations cause the computation of $d_{\mat{\beta}}$ is its raw form can take a significant amount of time compared to the estimation. In addition, the memory footprint is also reduced drastically.
Finally, for validating the estimation quality of the covariances $\mat{\Delta}_j$ the Frobenius norm of the ``true'' covariance $\mat{\Delta}=\bigotimes_{j = r}^1\mat{\Delta}_j$ to its estimate $\widehat{\mat{\Delta}}=\bigotimes_{j = r}^1\widehat{\mat{\Delta}}_j$ is computed.
There are also cases where the ``true'' covariance $\mat{\Delta}$ is \emph{not} a Kronecker product, in this situation the distance needs to be computed directly as $\|\mat{\Delta}-\widehat{\mat{\Delta}}\|_F$.
% % The first example (which by it self is \emph{not} exemplary) is the estimation with parameters $n = 200$, $p = 11$, $q = 5$, $k = 14$ and $r = 9$. The ``true'' matrices $\mat\alpha$, $\mat\beta$ generated by sampling there elements i.i.d. standard normal like the responses $y$. Then, for each observation, $\mat{f}_y$ is computed as $\sin(s_{i, j} y + o_{i j})$ \todo{ properly describe} to fill the elements of $\mat{f}_y$. Then the $\mat{X}$'s are samples as
% Lets considure a polynomial model with $r$ variables. Each variable by itself is described by a polynomial of order $p_j - 1$. Let $z_j$ be the variables for the $j$'th axis and let $\mat{\alpha}_j\in\mathbb{R}^{p_j}$, $j = 1, ..., r$ be the parameters for the polynomial of the $j$'th variable (note that the parameters $\mat{\alpha}_j$ are vectors treates by the estimation methods as $p_j\times 1$ matrices).
% For the simulation we take a 3D problem ($r = 3$) and quadratic polynomials. For $n = 500$ samples the parameters $\mat{z} = (z_1, z_2, z_3)$ are samples uniformly from the $[-1, 1]^3$ cube as arguments to quadratic polynomials $p_1 = p_2 = p_3 = 3$. The $i$'th scalar response $y_i$ is then computed by above formula which can be written in a multi-linear setting as
% where the order $r = 3$ tensor $\ten{X}$ consists of all interaction terms of the monoms $x_{j}^{k - 1}$ for $k = 1, ..., p_j$ for each axis $j = 1, ..., r$ ans $\sigma = 0.1$. More explicitly, the vectorized version satisfies
% where $\mat{\beta} = \bigotimes_{j = r}^1\mat{\alpha}_j$.
\subsection{EEG Data}
As an real world example we compair the HO-PIR method against two reference methods, namely LSIR \cite{lsir-PfeifferForzaniBura} and K-PIR \cite{sdr-PfeifferKaplaBura2021}, on the EEG data set. This is a small study including $77$ alcoholic individuals and $45$ control subjects (see: \url{http://kdd.ics.uci.edu/databases/eeg/eeg.data.html}). The data for each subject consisted of a $64\times256$ matrix, with each column representing a time point and each row a channel. The data were obtained by exposing each individual to visual stimuli and measuring $7$ Simulations voltage values from $64$ electrodes placed on the subjects scalps sampled at $256$ time points (at $256$ Hz for $1$ second). Each individual observation is the mean over $120$ different trial per subject. We first preprocess the data using the HO-PCA Algorithm~\ref{alg:HOPCA} with different PCA's per mode. The results are listed in Table~\ref{tab:eeg_sim}.
\caption{\label{tab:eeg_sim}Recreation of the EEG data LOO-CV from \cite{sdr-PfeifferKaplaBura2021} Section~7. The methods HO-PIR (ls) and HO-PIR (mle) use the ICU Algorithm~\ref{alg:iterative_updating}.}
% \caption{Mean (standard deviation) for simulated runs of $20$ repititions for the model $\mat{X} = \mat{\beta}\mat{f}_y\t{\mat{\alpha}}$ of dimensions $(p_1, p_2) = (11, 7)$, $(q_1, q_2) = (3, 5)$ with a sample size of $n = 200$. The covariance structure is $\mat{\Delta} = \mat{\Delta}_2\otimes \mat{\Delta}_1$ for $\Delta_i = \text{AR}(\sqrt{0.5})$, $i = 1, 2$. The functions applied to the standard normal response $y$ are $\sin, \cos$ with increasing frequency.}
% \caption{\label{tab:eeg_sim}Recreation of the EEG data LOO-CV from \cite{sdr-PfeifferKaplaBura2021} Section~7, EEG Data and Table~6 with new methods. Column \emph{vlp} stands for the Van Loan and Pitsianis initialization scheme as described in \cite{sdr-PfeifferKaplaBura2021} and \emph{ls} is the approach described above. The column \emph{npc} gives the number of principal component of the $(2D)^2 PCA$ preprocessing. Reduction by $\ten{X}\times_{j\in[r]}\t{\widehat{\mat{\alpha}}_j}$ instread of $^*$ where reduction is $\ten{X}\times_{j\in[r]}\t{\widehat{\mat{\Delta}}_j^{-1}\widehat{\mat{\alpha}}_j}$.}
Let $\mat A$ be a $p\times p$ dimensional non-singular matrix. Furthermore, let $\mat a, \mat b$ be $p$ vectors such that $\t{\mat b}A^{-1}\mat a\neq-1$, then
\section{Commutation Matrix and Permutation Identities}
\begin{center}
Note: In this section we use 0-indexing for the sake of simplicity!
\end{center}
In this section we summarize relations between the commutation matrix and corresponding permutation. We also list some extensions to ``simplify'' or represent some term. This is mostly intended for implementation purposes and understanding of terms occurring in the computations.
where $\pi_{p, q}$ is a permutation of the indices $i =0, ..., p q -1$ such that
\begin{displaymath}
\pi_{p, q}(i + j p) = j + i q, \quad\text{for }i = 0, ..., p - 1; j = 0, ..., q - 1.
\end{displaymath}
\begin{table}[!htp]
\centering
\begin{minipage}{0.8\textwidth}
\centering
\begin{tabular}{l c l}
$\mat{K}_{p, q}$&$\hat{=}$&$\pi_{p, q}(i + j p)= j + i q$\\
$\mat{I}_r\kron\mat{K}_{p, q}$&$\hat{=}$&$\tilde{\pi}_{p, q, r}(i + j p + k p q)= j + i q + k p q$\\
$\mat{K}_{p, q}\kron\mat{I}_r$&$\hat{=}$&$\hat{\pi}_{p, q, r}(i + j p + k p q)= r(j + i q)+ k$
\end{tabular}
\caption{\label{tab:commutation-permutation}Commutation matrix terms and corresponding permutations. Indices are all 0-indexed with the ranges; $i =0, ..., p -1$, $j =0, ..., q -1$ and $k =0, ..., r -1$.}
\end{minipage}
\end{table}
\section{Matrix and Tensor Operations}
The \emph{Kronecker product}\index{Operations!Kronecker@$\kron$ Kronecker product} is denoted as $\kron$ and the \emph{Hadamard product} uses the symbol $\circ$. We also need the \emph{Khatri-Rao product}\index{Operations!KhatriRao@$\hada$ Khatri-Rao product}
$\hada$ as well as the \emph{Transposed Khatri-Rao product}$\odot_t$ (or \emph{Face-Splitting product}). There is also the \emph{$n$-mode Tensor Matrix Product}\index{Operations!ttm@$\ttm[n]$$n$-mode tensor product} denoted by $\ttm[n]$ in conjunction with the \emph{$n$-mode Matricization} of a Tensor $\mat{T}$ written as $\mat{T}_{(n)}$, which is a matrix. See below for definitions and examples of these operations.\todo{ Definitions and Examples}
\todo{ resolve confusion between Khatri-Rao, Column-wise Kronecker / Khatri-Rao, Row-wise Kronecker / Khatri-Rao, Face-Splitting Product, .... Yes, its a mess.}
\paragraph{Kronecker Product $\kron$:}
\paragraph{Khatri-Rao Product $\hada$:}
\paragraph{Transposed Khatri-Rao Product $\odot_t$:} This is also known as the Face-Splitting Product and is the row-wise Kronecker product of two matrices. If relates to the Column-wise Kronecker Product through
\paragraph{$n$-mode unfolding:}\emph{Unfolding}, also known as \emph{flattening} or \emph{matricization}, is an reshaping of a tensor into a matrix with rearrangement of the elements such that mode $n$ corresponds to columns of the result matrix and all other modes are vectorized in the rows. Let $\ten{T}$ be a tensor of order $m$ with dimensions $t_1\times ... \times t_n\times ... \times t_m$ and elements indexed by $(i_1, ..., i_n, ..., i_m)$. The $n$-mode flattening, denoted $\ten{T}_{(n)}$, is defined as a $(t_n, \prod_{k\neq n}t_k)$ matrix with element indices $(i_n, j)$ such that $j =\sum_{k =1, k\neq n}^m i_k\prod_{l =1, l\neq n}^{k -1}t_l$.
\todo{ give an example!}
\paragraph{$n$-mode Tensor Product $\ttm[n]$:}
The \emph{$n$-mode tensor product}$\ttm[n]$ between a tensor $\mat{T}$ of order $m$ with dimensions $t_1\times t_2\times ... \times t_n\times ... \times t_m$ and a $p\times t_n$ matrix $\mat{M}$ is defined element-wise as
where $i_1, ..., i_{n-1}, i_{n+1}, ..., i_m$ run from $1$ to $t_1, ..., t_{n-1}, t_{n+1}, ..., t_m$, respectively. Furthermore, the $n$-th fiber index $j$ of the product ranges from $1$ to $p$. This gives a new tensor $\mat{T}\ttm[n]\mat{M}$ of order $m$ with dimensions $t_1\times t_2\times ... \times p\times ... \times t_m$.
\begin{example}[Matrix Multiplication Analogs]
Let $\mat{A}$, $\mat{B}$ be two matrices with dimensions $t_1\times t_2$ and $p\times q$, respectively. Then $\mat{A}$ is also a tensor of order $2$, now the $1$-mode and $2$-mode products are element wise given by
In other words, the $1$-mode product equals $\mat{A}\ttm[1]\mat{B}=\mat{B}\mat{A}$ and the $2$-mode is $\mat{A}\ttm[2]\mat{B}=\t{(\mat{B}\t{\mat{A}})}$ in the case of the tensor $\mat{A}$ being a matrix.
\end{example}
\begin{example}[Order Three Analogs]
Let $\mat{A}$ be a tensor of the form $t_1\times t_2\times t_3$ and $\mat{B}$ a matrix of dimensions $p\times q$, then the $n$-mode products have the following look
\begin{align*}
(\mat{A}\ttm[1]\mat{B})_{i,j,k}&= \sum_{l = 1}^{t_1}\mat{A}_{l,j,k}\mat{B}_{i,l}&\text{for }t_1 = q, \\
(\mat{A}\ttm[2]\mat{B})_{i,j,k}&= \sum_{l = 1}^{t_2}\mat{A}_{i,l,k}\mat{B}_{j,l}\equiv (\mat{B}\mat{A}_{i,:,:})_{j,k}&\text{for }t_2 = q, \\
or in other words, the $i$-th slice of the tensor product $\ten{F}\ttm[2]\mat{\beta}\ttm[3]\mat{\alpha}$ contains $\mat{\beta}\mat{f}_{y_i}\t{\mat{\alpha}}$ for $i =1, ..., n$.
In this section we give a short summary of alternative but equivalent operations.
Using the notation $\widehat{=}$ to indicate that two expressions are identical in the sense that they contain the same element in the same order but may have different dimensions. Meaning, when vectorizing ether side of $\widehat{=}$, they are equal ($\mat{A}\widehat{=}\mat{B}\ :\Leftrightarrow\ \vec{\mat{A}}=\vec{\mat{B}}$).
Therefore, we use $\mat{A}, \mat{B}, \mat{X}, \mat{F}, \mat{R}, ...$ for matrices. 3-Tensors are written as $\ten{A}, \ten{B}, \ten{T}, \ten{X}, \ten{F}, \ten{R}, ...$.
% if and only if $\vec\mat{X}\sim\mathcal{N}_{p q}(\vec\mat\mu, \mat\Delta_1\otimes\mat\Delta_2)$. Note the order of the covariance matrices $\mat\Delta_1, \mat\Delta_2$. Its density is given by
% \begin{displaymath}
% f(\mat{X}) = \frac{1}{(2\pi)^{p q / 2}|\mat\Delta_1|^{p / 2}|\mat\Delta_2|^{q / 2}}\exp\left(-\frac{1}{2}\tr(\mat\Delta_1^{-1}\t{(\mat X - \mat \mu)}\mat\Delta_2^{-1}(\mat X - \mat \mu))\right).
% \end{displaymath}
% \section{Sampling form a Multi-Array Normal Distribution}
% Let $\ten{X}$ be an order (rank) $r$ Multi-Array random variable of dimensions $p_1\times...\times p_r$ following a Multi-Array (or Tensor) Normal distributed
which means that $\E\widetilde{\mat\Delta}_1=\mat\Delta_1\tr(\mat\Delta_2)$ and in analogy $\E\widetilde{\mat\Delta}_2=\mat\Delta_2\tr(\mat\Delta_1)$. Now, we need to handle the scaling which can be estimated unbiasedly by
because with $\|\mat{R}_i\|_F^2=\tr\mat{R}_i\t{\mat{R}_i}=\tr\t{\mat{R}_i}\mat{R}_i$ the scale estimate $\tilde{s}=\tr(\widetilde{\mat\Delta}_1)=\tr(\widetilde{\mat\Delta}_2)$. Then $\E\tilde{s}=\tr(\E\widetilde{\mat\Delta}_1)=\tr{\mat\Delta}_1\tr{\mat\Delta}_2=\tr({\mat\Delta}_1\otimes{\mat\Delta}_2)$. Leading to the estimate of the covariance as
\todo{ prove they are consistent, especially $\widetilde{\mat\Delta}=\tilde{s}^{-1}(\widetilde{\mat\Delta}_1\otimes\widetilde{\mat\Delta}_2)$!}
The hoped for a benefit is that these covariance estimates are in a closed form which means there is no need for an additional iterative estimations step. Before we start with the derivation of the gradients define the following two quantities
Now, the matrix normal with the covariance matrix of the vectorized quantities of the form $\mat{\Delta}= s^{-1}(\mat{\Delta}_1\otimes\mat{\Delta}_2)$ has the form
The second form is due to the property of the determinant for scaling and the Kronecker product giving that $|\widetilde{\mat\Delta}| =(\tilde{s}^{-1})^{p q}|\widetilde{\mat{\Delta}}_1|^p |\widetilde{\mat{\Delta}}_2|^q$ as well as an analog Kronecker decomposition as in the MLE case.
The last one is tedious but straight forward. Its computation extensively uses the symmetry of $\widetilde{\mat{\Delta}}_1$, $\widetilde{\mat{\Delta}}_2$, the cyclic property of the trace and the relation $\d\mat{A}^{-1}=-\mat{A}^{-1}(\d\mat{A})\mat{A}^{-1}$.
Putting it all together
\begin{align*}
\d\tilde{l}(\mat{\alpha}, \mat{\beta})
&= \frac{n p q}{2}\Big(-\frac{2}{n\tilde{s}}\Big)\sum_{i = 1}^n \tr(\t{\mat{f}_{y_i}}\t{\mat{\beta}}\mat{R}_i\d\mat{\alpha} + \mat{f}_{y_i}\t{\mat{\alpha}}\t{\mat{R}_i}\d\mat{\beta}) \\
Observe that the bracketed expressions before $\d\mat{\alpha}$ and $\d\mat{\beta}$ are transposes. Lets denote the expression for $\d\mat{\alpha}$ as $\mat{G}_i$ which has the form
\begin{displaymath}
\mat{G}_i
= (\tr(\widetilde{\mat{\Delta}}_1^{-1}\mat{S}_1) - p q \tilde{s}^{-1})\mat{R}_i
This section contains short summaries of the main references with each sub-section concerning one paper.
\subsection{}
\subsection{Generalized Tensor Decomposition With Features on Multiple Modes}
The \cite{TensorDecomp-HuLeeWang2022} paper proposes a multi-linear conditional mean model for a constraint rank tensor decomposition. Let the responses $\ten{Y}\in\mathbb{R}^{d_1\times ... \times\d_K}$ be an order $K$ tensor. Associated with each mode $k\in[K]$ they assume feature matrices $\mat{X}_k\in\mathbb{R}^{d_k\times p_k}$. Now, they assume that conditional on the feature matrices $\mat{X}_k$ the entries of the tensor $\ten{Y}$ are independent realizations. The rank constraint is specified through $\mat{r}=(r_1, ..., r_K)$, then the model is given by
The order $K$ tensor $\ten{C}\in\mathbb{R}^{r_1\times...\times r_K}$ is an unknown full-rank core tensor and the matrices $\mat{M}_k\in\mathbb{R}^{p_k\times r_k}$ are unknown factor matrices. The function $f$ is applied element wise and serves as the link function based on the assumed distribution family of the tensor entries. Finally, the operation $\times$ denotes the tensor-by-matrix product using a short hand
with $\ttm[k]$ denoting the $k$-mode tensor matrix product.
The algorithm for estimation of $\ten{C}$ and $\mat{M}_1, ..., \mat{M}_K$ assumes the individual conditional entries of $\ten{Y}$ to be independent and to follow a generalized linear model with link function $f$. The proposed algorithm is an iterative algorithm for minimizing the negative log-likelihood
where $b = f'$ it the derivative of the canonical link function $f$ in the generalized linear model the conditioned entries of $\ten{Y}$ follow. The algorithm utilizes the higher-order SVD (HOSVD) to enforce the rank-constraint.
The main benefit is that this approach generalizes well to a multitude of different structured data sets.
\todo{ how does this relate to the $\mat{X}=\mat{\mu}+\mat{\beta}\mat{f}_y\t{\mat{\alpha}}+\mat{\epsilon}$ model.}
