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effie review of tensor normal MLE estimation

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Daniel Kapla 2023-12-01 13:49:34 +01:00
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@ -479,9 +479,7 @@ where $\mat{B} = \bigotimes_{j = r}^{1}\mat{\beta}_j$ and the component matrices
\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} \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} \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 $c$. 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 will relax the assumptions for $\mat{\beta}_k$ and $\mat{\Omega}_k$ as a consequence of \cref{thm:param-manifold}, which is subject of \cref{sec:kron-manifolds,sec:matrix-manifolds}. 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}.
% \todo{Maybe already here introduce the ``constraint'' set of $\Omega$'s allowed as $\{ \Omega\in\mathbb{R}_{++}^{p\times p} : \vec{\Omega} = \t{(\pinv{(\mat{T}_2\pinv{\mat{D}_p})}\mat{T}_2\pinv{\mat{D}_p})}\vec{\Omega} \}$}
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 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} \begin{align}
@ -548,10 +546,10 @@ The maximum likelihood estimate of $\mat{\theta}_0$ is the solution to the optim
\end{equation} \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 $\widehat{\mat{\Omega}} = \bigkron_{k = r}^{1}\widehat{\mat{\Omega}}_k$. 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}. For applying gradient based optimization, we compute the gradients of $l_n$ in \cref{thm:grad}. 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} \begin{theorem}\label{thm:grad}
For $n$ i.i.d. observations $(\ten{X}_i, y_i), i = 1, ..., n$ the log-likelihood has the form \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 $k = 1, ..., r$. Furthermore, let $\ten{G}_2(\mat{\eta}_y)$ be a tensor of dimensions $p_1, \ldots, p_r$ such that 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} \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)}. \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} \end{displaymath}
@ -561,49 +559,47 @@ A straightforward and general method for parameter estimation is \emph{gradient
\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{\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})} \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*} \end{align*}
giving the gradient $\nabla l_n = (\nabla_{\overline{\ten{\eta}}}l_n, \nabla_{\mat{\beta}_1}l_n, \ldots, \nabla_{\mat{\beta}_r}l_n, \nabla_{\mat{\Omega}_1}l_n, \ldots, \nabla_{\mat{\Omega}_r}l_n)$. 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 $\ten{G}_2(\mat{\eta}_y) = \ten{g}_2(\mat{\eta}_y)$.
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} \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 case of the tensor normal distribution an iterative cyclic updating scheme is derived in \cref{sec:tensor_normal_estimation}. This method has much faster convergence, is stabel and does not requires any hyper parameters. 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}. 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} \subsection{Tensor Normal}\label{sec:tensor_normal_estimation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
With $\ten{X}\mid Y = y$ following a tensor normal distribution, its density, with mean $\ten{\mu}_y$ and covariance $\mat{\Sigma} = \bigkron_{k = r}^{1}\mat{\Sigma}_k$ is given by 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} \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). 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} \end{displaymath}
We assume the distribution is non-degenerate which means that the covariances $\mat{\Sigma}_k$ are symmetric positive definite matrices. For the sake of simplicity, we w.l.o.g. assume $\ten{X}$ to have marginal expectation equal zero, $\E\ten{X} = 0$. Rewriting this in the quadratic exponential family form \eqref{eq:quadratic-exp-fam}, determines the scaling constant $c = -1/2$, and give the relation to the GMLM parameters $\overline{\ten{\eta}}, \mat{\beta}_k$ and $\mat{\Omega}_k$, for $k = 1, \ldots, r$ as 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 $\overline{\ten{\eta}}, \mat{\beta}_k$ and $\mat{\Omega}_k$, for $k = 1, \ldots, r$ is
\begin{displaymath} \begin{displaymath}
\ten{\mu}_y = \ten{F}_y\mlm_{k = 1}^{r}\mat{\Omega}_k^{-1}\mat{\beta}_k, \qquad \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} \mat{\Omega}_k = \mat{\Sigma}_k^{-1},
\end{displaymath} \end{displaymath}
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 the matrix $\mat{T}_2$ from \eqref{eq:t-stat} is equal to the identity. This also means that the gradients of the log-likelihood $l_n$ in \cref{thm:grad} are simpler as well. We get 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} \begin{displaymath}
\ten{g}_1(\mat{\eta}_y) = \E[\ten{X}\mid Y = y] = \ten{\mu}_y, \qquad \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)}. \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} \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
In practice, we start the estimation process by demeaning a given data set $(\ten{X}_i, \ten{F}_{y_i})$ with $i = 1, \ldots, n$ observations. After demenaing, only the reduction matrices $\mat{\beta}_k$ and the scatter matrices $\mat{\Omega}_k$ need to be estimated. Then, to solve the optimization problem \eqref{eq:mle}, with $\overline{\ten{\eta}} = 0$ after demeaning, we need some initial values. For initial estimates $\hat{\mat{\beta}}_k^{(0)}$ we use a simple heuristic approach. First, we compute moment based mode-wise marginal covariance estimates $\widehat{\mat{\Sigma}}_k(\ten{X})$ and $\widehat{\mat{\Sigma}}_k(\ten{F}_Y)$ given by
\begin{displaymath} \begin{displaymath}
\widehat{\mat{\Sigma}}_k(\ten{X}) = \frac{1}{n}\sum_{i = 1}^{n} (\ten{X}_i)_{(k)}\t{(\ten{X}_i)_{(k)}}, \qquad \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)}}. \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} \end{displaymath}
Then, for every mode $k = 1, \ldots, r$, we compute the first $j = 1, \ldots, q_k$ eigen vectors $\mat{v}_j(\widehat{\mat{\Sigma}}_k(\ten{X}))$, $\mat{v}_j(\widehat{\mat{\Sigma}}_k(\ten{F}_Y))$ and eigen values $\lambda_j(\widehat{\mat{\Sigma}}_k(\ten{X}))$, $\lambda_j(\widehat{\mat{\Sigma}}_k(\ten{X}))$ of the marginal covariance estimates. Let, 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*} \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{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{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)). \\ \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*} \end{align*}
The initial estimates are then set to The initial value of $\mat{\beta}_k$ is
\begin{displaymath} \begin{displaymath}
\hat{\mat{\beta}}_k^{(0)} = \mat{U}_k\sqrt{\mat{D}_k}\t{\mat{V}_k}. \hat{\mat{\beta}}_k^{(0)} = \mat{U}_k\sqrt{\mat{D}_k}\t{\mat{V}_k},
\end{displaymath} \end{displaymath}
The initial values of $\mat{\Omega}_k$ are set to the identity $\mat{\Omega}_k^{(0)} = \mat{I}_{p_k}$. 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 any estimates $\hat{\mat{\beta}}_1, \ldots, \hat{\mat{\beta}}_r, \hat{\mat{\Omega}}_1, \ldots, \hat{\mat{\Omega}}_r$, we take the gradient $\nabla_{\mat{\beta}_j}l_n$ of the log-likelihood $l_n$ in \eqref{eq:log-likelihood} of the tensor normal distributed from \cref{thm:grad} and keep all other parameters except $\mat{\beta}_j$ fixed. Then, $\nabla_{\mat{\beta}_j}l_n = 0$ has a closed form solution 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} \begin{equation}\label{eq:tensor_normal_beta_solution}
\t{\mat{\beta}_j} = \biggl( \t{\mat{\beta}_j} = \biggl(
\sum_{i = 1}^{n} \sum_{i = 1}^{n}
@ -617,29 +613,28 @@ Given any estimates $\hat{\mat{\beta}}_1, \ldots, \hat{\mat{\beta}}_r, \hat{\mat
\biggr) \biggr)
\hat{\mat{\Omega}}_j. \hat{\mat{\Omega}}_j.
\end{equation} \end{equation}
For the scatter matrices $\mat{\Omega}_j$, we need to fudge a bit. 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 practive thoug, it is more stable, faster and equaly accurate to use mode-wise covariance estimates via the residuals 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} \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. \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} \end{displaymath}
The estimates are computed via the inbetween values The estimates are computed via
\begin{displaymath} \begin{displaymath}
\tilde{\mat{\Sigma}}_j = \sum_{i = 1}^n (\hat{\ten{R}}_i)_{(j)} \t{(\hat{\ten{R}}_i)_{(j)}} \tilde{\mat{\Sigma}}_j = \sum_{i = 1}^n (\hat{\ten{R}}_i)_{(j)} \t{(\hat{\ten{R}}_i)_{(j)}},
\end{displaymath} \end{displaymath}
which only lacks scaling $\tilde{s}\tilde{\mat{\Sigma}}_j = \hat{\mat{\Omega}}_j^{-1}$. The scaling condition is that the mean squared error has to be equal to the trace of the covariance estimate, 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} \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}. \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} \end{displaymath}
Solving for the scaling value gives so that
\begin{displaymath} \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} \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} \end{displaymath}
resulting in the estimates $\hat{\mat{\Omega}}_j = (\tilde{s}\tilde{\mat{\Sigma}}_j)^{-1}$. 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.
Estimation is then performed by updating for $j = 1, \ldots, r$ the estimates $\hat{\mat{\beta}}_j$ via \eqref{eq:tensor_normal_beta_solution}, then we recompute the $\hat{\mat{\Omega}}_j$ estimates simultaniously keeping the $\hat{\mat{\beta}}_j$'s fixed. This procedure is repeated untill 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.
A technical detail for numerical stability is to ensure that the scaled inbetween values $\tilde{s}\tilde{\mat{\Sigma}}_j$, assumed to be symmetric and positive definite, are well conditioned. Otherwise, the procedure can be numerically unstable by attempting to compute $\hat{\mat{\Omega}}_j = (\tilde{s}\tilde{\mat{\Sigma}}_j)^{-1}$. Therefore, we estimate the condition number of $\tilde{s}\tilde{\mat{\Sigma}}_j$ before computing the inverse. In case of being ill conditioned, we regularize and use $\hat{\mat{\Omega}}_j = (\tilde{s}\tilde{\mat{\Sigma}}_j + 0.2 \lambda_{1}(\tilde{s}\tilde{\mat{\Sigma}}_j)\mat{I}_{p_j})^{-1}$ instead, where $\lambda_{1}(\tilde{s}\tilde{\mat{\Sigma}}_j)$ is the first (maximum) eigen value. 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.
Furthermore, we may assume that the parameter space follows a more generel setting as in \cref{thm:param-manifold}. In this case, updating may produces estimates outside of the parameter space. A simple and efficient method is to project every updated estimate onto the corresponding manifold.
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\subsection{Ising Model}\label{sec:ising_estimation} \subsection{Ising Model}\label{sec:ising_estimation}
@ -664,7 +659,7 @@ We extend the conditional PMF by allowing the binary variables to be tensor valu
\begin{align} \begin{align}
P_{\mat{\gamma}_y}(\ten{X} \mid Y = y) 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(\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) &= 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} \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 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. 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 where we set $\overline{\ten{\eta}} = 0$ and $\mat{T}_2$ to the identity. This is an additional constraint to the model, the reason 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. 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} \begin{equation}\label{eq:ising-natural-params}
@ -744,7 +739,7 @@ For estimation of dimensions $p$ bigger than $20$, we use a Monte-Carlo method t
\subsection{Kronecker Product Manifolds}\label{sec:kron-manifolds} \subsection{Kronecker Product Manifolds}\label{sec:kron-manifolds}
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\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. \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. 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.