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GNU GENERAL PUBLIC LICENSE




Version 3, 29 June 2007




Copyright © 2007 Free Software Foundation, Inc. <http s ://fsf.org/>




Everyone is permitted to copy and distribute verbatim copies of this license


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Preamble




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17. Interpretation of Sections 15 and 16.




If the disclaimer of warranty and limitation of liability provided above cannot


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TERMS AND CONDITIONS




How to Apply These Terms to Your New Programs




If you develop a new program, and you want it to be of the greatest possible


use to the public, the best way to achieve this is to make it free software


which everyone can redistribute and change under these terms.




To do so, attach the following notices to the program. It is safest to attach


them to the start of each source file to most effectively state the exclusion


of warranty; and each file should have at least the "copyright" line and a


pointer to where the full notice is found.




<one line to give the program's name and a brief idea of what it does.>




Copyright (C) <year> <name of author>




This program is free software: you can redistribute it and/or modify it under


the terms of the GNU General Public License as published by the Free Software


Foundation, either version 3 of the License, or (at your option) any later


version.




This program is distributed in the hope that it will be useful, but WITHOUT


ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS


FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.




You should have received a copy of the GNU General Public License along with


this program. If not, see <http s ://www.gnu.org/licenses/>.




Also add information on how to contact you by electronic and paper mail.




If the program does terminal interaction, make it output a short notice like


this when it starts in an interactive mode:




<program> Copyright (C) <year> <name of author>




This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.




This is free software, and you are welcome to redistribute it under certain


conditions; type `show c' for details.




The hypothetical commands `show w' and `show c' should show the appropriate


parts of the General Public License. Of course, your program's commands might


be different; for a GUI interface, you would use an "about box".




You should also get your employer (if you work as a programmer) or school,


if any, to sign a "copyright disclaimer" for the program, if necessary. For


more information on this, and how to apply and follow the GNU GPL, see <http


s ://www.gnu.org/licenses/>.




The GNU General Public License does not permit incorporating your program


into proprietary programs. If your program is a subroutine library, you may


consider it more useful to permit linking proprietary applications with the


library. If this is what you want to do, use the GNU Lesser General Public


License instead of this License. But first, please read <http s ://www.gnu.org/


licenses /whynotlgpl.html>.


@ 0,0 +1,504 @@


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pdfauthor = {Daniel Kapla},


pdfcreator = {\pdftexbanner}


}


}




\makeindex




% Bibliography resource(s)


\addbibresource{main.bib}




% Setup environments


% Theorem, Lemma


\theoremstyle{plain}


\newtheorem{theorem}{Theorem}


\newtheorem{lemma}{Lemma}


\newtheorem{example}{Example}


% Definition


\theoremstyle{definition}


\newtheorem{defn}{Definition}


% Remark


\theoremstyle{remark}


\newtheorem{remark}{Remark}




% Define math macros


\newcommand{\mat}[1]{\boldsymbol{#1}}


\newcommand{\ten}[1]{\mathcal{#1}}


\renewcommand{\vec}{\operatorname{vec}}


\newcommand{\unvec}{\operatorname{vec^{1}}}


\newcommand{\reshape}[1]{\operatorname{reshape}_{#1}}


\newcommand{\vech}{\operatorname{vech}}


\newcommand{\rank}{\operatorname{rank}}


\newcommand{\diag}{\operatorname{diag}}


\DeclareMathOperator{\tr}{tr}


\DeclareMathOperator{\var}{Var}


\DeclareMathOperator{\cov}{Cov}


\DeclareMathOperator{\Span}{Span}


\DeclareMathOperator{\E}{\operatorname{\mathbb{E}}}


% \DeclareMathOperator{\independent}{{\bot\!\!\!\bot}}


\DeclareMathOperator*{\argmin}{{arg\,min}}


\DeclareMathOperator*{\argmax}{{arg\,max}}


\newcommand{\D}{\textnormal{D}} % derivative


\renewcommand{\d}{\textnormal{d}} % differential


\renewcommand{\t}[1]{{#1^{\prime}}} % matrix transpose


\newcommand{\pinv}[1]{{#1^{\dagger}}} % `MoorePenrose pseudoinverse`


\newcommand{\invlink}{\widetilde{\mat{g}}}




\newcommand{\todo}[1]{{\color{red}TODO: #1}}


\newcommand{\effie}[1]{{\color{blue}Effie: #1}}




% Pseudo Code Commands


\newcommand{\algorithmicbreak}{\textbf{break}}


\newcommand{\Break}{\State \algorithmicbreak}




\begin{document}




\maketitle




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%% Abstract %%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{abstract}


We propose a method for sufficient dimension reduction of Tensorvalued predictor (multi dimensional arrays) for regression or classification. We assume an Quadratic Exponential Family for a Generalized Linear Model in an inverse regression setting where the relation via a link is of a multilinear nature.


Using a multilinear relation allows to perform peraxis reductions which reduces the total number of parameters drastically for higher order Tensorvalued predictors. Under the Exponential Family we derive maximum likelihood estimates for the multilinear sufficient dimension reduction of the Tensorvalued predictors. Furthermore, we provide an estimation algorithm which utilizes the Tensor structure allowing efficient implementations. The performance of the method is illustrated via simulations and real world examples are provided.


\end{abstract}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Quadratic Exponential Family GLM}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\begin{description}


\item[Distribution]


\begin{displaymath}


f_{\mat{\theta}_y}(\ten{X}\mid Y = y) = h(\ten{X})\exp(\t{\mat{\eta}(\mat{\theta}_y)}\mat{t}(\ten{X})  b(\mat{\theta}_y))


\end{displaymath}


\item[(inverse) link]


\begin{displaymath}


\invlink(\mat{\eta}(\mat{\theta}_y)) = \E_{\mat{\theta}_y}[\mat{t}(\ten{X})\mid Y = y]


\end{displaymath}


\item[(multi) linear predictor] For


\begin{displaymath}


\mat{\eta}_y = \mat{\eta}(\mat{\theta}_y) = \begin{pmatrix}


\mat{\eta}_1(\mat{\theta}_y) \\


\mat{\eta}_2(\mat{\theta}_y)


\end{pmatrix},\qquad


\mat{t}(\ten{X}) = \begin{pmatrix}


\mat{t}_1(\ten{X}) \\


\mat{t}_2(\ten{X})


\end{pmatrix} = \begin{pmatrix}


\vec{\ten{X}} \\


\vec{\ten{X}}\otimes\vec{\ten{X}}


\end{pmatrix}


\end{displaymath}


where


\begin{align*}


\mat{\eta}_1(\mat{\theta}_y) &= \mat{\eta}_{y,1} = c_1 \vec(\overline{\ten{\eta}}_1 + \ten{F}_y\times_{k\in[r]}\mat{\alpha}_k) \\


\mat{\eta}_2(\mat{\theta}_y) &= \mat{\eta}_{y,2} = c_2 \vec{\bigotimes_{k = r}^1 \mat{\Omega}_k}


\end{align*}


with model parameters $\overline{\ten{\eta}}_1, \mat{\alpha}_1, ..., \mat{\alpha}_r, \mat{\Omega}_1, ..., \mat{\Omega}_r$ where $\overline{\ten{\eta}}_1$ is a $p_1\times ... \times p_r$ tensor, $\mat{\alpha}_j$ are $p_j\times q_j$ unconstrained matrices and $\mat{\Omega}_j$ are symmetric $p_j\times p_j$ matrices for each of the $j = 1, ..., r$ modes. Finally, $c_1$ and $c_2$ are known constants simplifying modeling for specific distributions.


\end{description}


% With that approach we get


% \begin{displaymath}


% \t{\mat{\eta}(\mat{\theta}_y)}\mat{t}(\ten{X}) = \t{\mat{\eta}_{y,1}}\mat{t}_1(\ten{X}) + \t{\mat{\eta}_{y,2}}\mat{t}_2(\ten{X}) = \langle\overline{\ten{\eta}}_1 + \ten{F}_y\times_{k\in[r]}\mat{\alpha}_k, \ten{X} \rangle + \langle\ten{X}\times_{k\in[r]}\mat{\Omega}_k, \ten{X} \rangle.


% \end{displaymath}




\begin{theorem}[LogLikelihood and Score]


For $n$ i.i.d. observations $(\ten{X}_i, y_i), i = 1, ..., n$ the loglikelihood has the form


\begin{displaymath}


l(\mat{\eta}_y) = \sum_{i = 1}^n(\log h(\ten{X}_i) + c_1\langle\overline{\ten{\eta}}_1 + \ten{F}_{y_i}\times_{k\in[r]}\mat{\alpha}_k, \ten{X}_i \rangle + c_2\langle\ten{X}_i\times_{k\in[r]}\mat{\Omega}_k, \ten{X}_i \rangle  b(\mat{\eta}_{y_i})).


\end{displaymath}


% The MLE estimate for the intercept term $\overline{\ten{\eta}}_1$ is


% \begin{displaymath}


% \widehat{\ten{\eta}}_1 = \frac{1}{n}\sum_{i = 1}^n \ten{X}_i


% \end{displaymath}


The gradients with respect to the GLM parameters $\overline{\ten{\eta}}_1$, $\mat{\alpha}_j$ and $\mat{\Omega}_j$ for $j = 1, ..., r$ are given by


\begin{align*}


\nabla_{\overline{\ten{\eta}}_1}l &= c_1\sum_{i = 1}^n \reshape{\mat{p}}(\mat{t}_1(\ten{X}_i)  \invlink_1(\mat{\eta}_{y_i})), \\


\nabla_{\mat{\alpha}_j}l &= c_1 \sum_{i = 1}^n \reshape{\mat{p}}(\mat{t}_1(\ten{X}_i)  \invlink_1(\mat{\eta}_{y_i}))_{(j)}\t{(\ten{F}_{y_i}\times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}}, \\


\vec\nabla_{\mat{\Omega}_j}l &= c_2 \mat{D}_{p_j}\t{\mat{D}_{p_j}} \reshape{(\mat{p}, \mat{p})}\!\!\Big(\sum_{i = 1}^n(\mat{t}_2(\ten{X}_i)  \invlink_2(\mat{\eta}_{y_i}))\Big)_{(j, r + j)}\vec\bigotimes_{\substack{k = r\\k\neq j}}^{1}\mat{\Omega}_k


\end{align*}


% The Fisher Information for the GLM parameters is given block wise by


% \begin{displaymath}


% % \mathcal{I}_{\ten{X}\mid Y = y}(\vec{\overline{\ten{\eta}}_1}, \vec\mat{\alpha}_1, ..., \vec\mat{\alpha}_r, \vec\mat{\Omega}_1, ..., \vec\mat{\Omega}_r) = \begin{pmatrix}


% \mathcal{I}_{\ten{X}\mid Y = y} = \begin{pmatrix}


% \mathcal{I}(\overline{\ten{\eta}}_1) & \mathcal{I}(\overline{\ten{\eta}}_1, \mat{\alpha}_1) & \cdots & \mathcal{I}(\overline{\ten{\eta}}_1, \mat{\alpha}_r) & \mathcal{I}(\overline{\ten{\eta}}_1, \mat{\Omega}_1) & \cdots & \mathcal{I}(\overline{\ten{\eta}}_1, \mat{\Omega}_r) \\


% \mathcal{I}(\mat{\alpha}_1, \overline{\ten{\eta}}_1) & \mathcal{I}(\mat{\alpha}_1) & \cdots & \mathcal{I}(\mat{\alpha}_1, \mat{\alpha}_r) & \mathcal{I}(\mat{\alpha}_1, \mat{\Omega}_1) & \cdots & \mathcal{I}(\mat{\alpha}_1, \mat{\Omega}_r) \\


% \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\


% \mathcal{I}(\mat{\alpha}_r, \overline{\ten{\eta}}_1) & \mathcal{I}(\mat{\alpha}_r, \mat{\alpha}_1) & \cdots & \mathcal{I}(\mat{\alpha}_r) & \mathcal{I}(\mat{\alpha}_r, \mat{\Omega}_1) & \cdots & \mathcal{I}(\mat{\alpha}_r, \mat{\Omega}_r) \\


% \mathcal{I}(\mat{\Omega}_1, \overline{\ten{\eta}}_1) & \mathcal{I}(\mat{\Omega}_1, \mat{\alpha}_1) & \cdots & \mathcal{I}(\mat{\Omega}_1, \mat{\alpha}_r) & \mathcal{I}(\mat{\Omega}_1) & \cdots & \mathcal{I}(\mat{\Omega}_1, \mat{\Omega}_r) \\


% \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\


% \mathcal{I}(\mat{\Omega}_r, \overline{\ten{\eta}}_1) & \mathcal{I}(\mat{\Omega}_r, \mat{\alpha}_1) & \cdots & \mathcal{I}(\mat{\Omega}_r) & \mathcal{I}(\mat{\Omega}_r, \mat{\Omega}_1) & \cdots & \mathcal{I}(\mat{\Omega}_r)


% \end{pmatrix}


% \end{displaymath}


% where


% \begin{align*}


% \mathcal{I}(\overline{\ten{\eta}}_1) &= \sum_{i = 1}^n \cov_{\mat{\theta}_{y_i}}(\vec\ten{X}\mid Y = y_i), \\


% \mathcal{I}(\mat{\alpha}_j) &= \sum_{i = 1}^n ((\ten{F}_{y_i}\times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}\otimes\mat{I}_{p_j})\mat{K}_{\mat{p},(j)}\cov_{\mat{\theta}_{y_i}}(\vec\ten{X}\mid Y = y_i)\t{\mat{K}_{\mat{p},(j)}}(\t{(\ten{F}_{y_i}\times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}}\otimes\mat{I}_{p_j}), \\


% \mathcal{I}(\mat{\alpha}_j) &= \sum_{i = 1}^n \todo{continue}


% \end{align*}




% \todo{Fisher Information}


\end{theorem}




Illustration of dimensions


\begin{displaymath}


\underbrace{ \mat{D}_{p_j}\t{\mat{D}_{p_j}} }_{\makebox[0pt]{\scriptsize $p_j^2\times p_j^2$}}


%


\underbrace{%


\overbrace{\reshape{(\mat{p}, \mat{p})}\!\!\Big(\sum_{i = 1}^n


\underbrace{ (\mat{t}_2(\ten{X}_i)  \invlink_2(\mat{\eta}_{y_i}) }_{p^2\times 1}


\Big)}^{\substack{\text{(tensor of order $2 r$)}\\p_1\times p_2\times ... \times p_r\times p_1\times p_2\times ... \times p_r}} \!\!\makebox[0pt]{\phantom{\Big)}}_{(j, r + j)}


}_{\substack{p_j^2\times (p / p_j)^2\\\text{(matricized / put $j$ mode axis to the front)}}}


%


\underbrace{%


\vec \overbrace{ \bigotimes_{\substack{k = r\\k\neq j}}^{1}\mat{\Omega}_j }^{\makebox[0pt]{\scriptsize $(p/p_j)\times (p/p_j)$}}


}_{\makebox[0pt]{\scriptsize $(p/p_j)^2\times 1$}}


\end{displaymath}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Sufficient Dimension Reduction}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%






\begin{theorem}[SDR]\label{thm:sdr}


A sufficient reduction for the regression $y\mid \ten{X}$ under the quadratic exponential family inverse regression model \todo{reg} is given by


\begin{displaymath}


R(\ten{X}) = \vec(\ten{X}\times_{k\in[r]}\mat{\Omega}_k\mat{\alpha}_k).


\end{displaymath}


\todo{type proof in appendix}


\end{theorem}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Special Distributions}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


We illustrate the SDR method on two special cases, first the Tensor Normal distribution and second on the MultiVariate Bernoulli distribution with vector, matrix and tensor valued predictors.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Tensor Normal}




Let $\ten{X}, \ten{F}_y$ be order $r$ tensors of dimensions $p_1\times ... \times p_r$ and $q_1\times ... \times q_r$, respectively. We assume the inverse regression model for $\ten{X}\mid Y = y$ to be tensor normal distributed with density


\begin{displaymath}


f_{\mat{\theta}_y}(\ten{X}\mid Y = y) = (2\pi)^{p/2}\prod_{k = 1}^r \mat{\Delta}_{k}^{p / 2 p_{k}}\exp\Big(


\frac{1}{2}\langle \ten{X}  \ten{\mu}_y, (\ten{X}  \ten{\mu}_y)\times_{k\in[r]}\mat{\Delta}_{k}^{1} \rangle


\Big)


\end{displaymath}


with location parameter tensor $\ten{\mu}_y$ depending on $y$ and the symmetric covariance matrices $\mat{\Delta}_{k}$ for each of the $k\in[r]$ modes (independent of $y$) collected in the parameter vector $\mat{\theta}_y = (\ten{\mu}_y, \mat{\Delta}_1, ..., \mat{\Delta}_r)$. Rewriting into the form of an quadratic exponential family leads to


\begin{align*}


f_{\mat{\theta}_y}(\ten{X}\mid Y = y)


&= (2\pi)^{p/2} \exp\Big(


\frac{1}{2}\langle \ten{X}, \ten{X}\times_{k\in[r]}\mat{\Delta}_{k}^{1} \rangle


+\langle \ten{X}, \ten{\mu}_y\times_{k\in[r]}\mat{\Delta}_k^{1} \rangle \\


&\makebox[10em]{}\frac{1}{2}\langle \ten{\mu}_y, \ten{\mu}_y\times_{k\in[r]}\mat{\Delta}_{k}^{1} \rangle


\sum_{k = 1}^r \frac{p}{2 p_{k}}\log\mat{\Delta}_k


\Big) \\


&= h(\ten{X})\exp(\t{\mat{{\eta}}(\mat{\theta}_y)}\mat{t}(\ten{X})  b(\mat{\theta}_y)).


\end{align*}


Identifying the exponential family components gives


\begin{align*}


h(\ten{X}) &= (2\pi)^{p/2} \\


b(\mat{\theta}_y) &= \frac{1}{2}\langle \ten{\mu}_y, \ten{\mu}_y\times_{k\in[r]}\mat{\Delta}_{k}^{1} \rangle + \sum_{k = 1}^r \frac{p}{2 p_{k}}\log\mat{\Delta}_{k}


\end{align*}


and


\begin{align*}


\mat{\eta}(\mat{\theta}_y) &= (\mat{\eta}_1(\mat{\theta}_y); \mat{\eta}_2(\mat{\theta}_y)) &


\mat{t}(\ten{X}) &= (\mat{t}_1(\ten{X}); \mat{t}_2(\ten{X}))


\end{align*}


where


\begin{align*}


\mat{\eta}_1(\mat{\theta}_y) = \mat{\eta}_{y,1} &= \vec(\ten{\mu}_y\times_{k\in[r]}\mat{\Delta}_{k}^{1}), &


\mat{t}_1(\ten{X}) &= \vec\ten{X}, \\


\mat{\eta}_2(\mat{\theta}_y) = \mat{\eta}_{y,2} &= \frac{1}{2}\vec\bigotimes_{k = r}^{1}\mat{\Delta}_{k}^{1}, &


\mat{t}_2(\ten{X}) &= \vec\ten{X}\otimes\vec\ten{X}.


\end{align*}


The natural parameters are models as described in the MultiLinear GLM as


\begin{align*}


\mat{\eta}_{y,1} &= \vec(\overline{\ten{\eta}}_1 + \ten{F}_y\times_{k\in[r]}\mat{\alpha}_{k}) \\


\mat{\eta}_{y,2} &= \frac{1}{2}\vec\bigotimes_{k = r}^{1}\mat{\Omega}_{k}.


\end{align*}


The intercept parameter $\overline{\ten{\eta}}_1$ is of the same dimensions as $\ten{X}$ and the reduction matrices $\mat{\alpha}_j$ are of dimensions $p_j\times q_j$ while the symmetric $\mat{\Omega}_j$ are of dimensions $p_j\times p_j$. The inverse relation from the GLM parameters to the tensor normal parameters is


\begin{align*}


\ten{\mu}_y &= (\overline{\ten{\eta}}_1 + \ten{F}_y\times_{j\in[r]}\mat{\alpha}_{j})\times_{k\in[r]}\mat{\Omega}_{k}^{1} = (\unvec(2\mat{\eta}_{y,2}))^{1}\mat{\eta}_{y,1} \\


\mat{\Delta}_{k} &= \mat{\Omega}_{k}^{1}


\end{align*}


for each $j\in[r]$. The inverse link is given by


\begin{displaymath}


\invlink(\mat{\eta}_y) = \E_{\mat{\theta}_y}[\mat{t}(\ten{X})\mid Y = y]


\end{displaymath}


consisting of the first and second (uncentered) vectorized moments of the tensor normal distribution.


\begin{align*}


\invlink_1(\mat{\eta}_y) &\equiv \E[\ten{X} \mid Y = y] = \ten{\mu}_y \\


&= (\ten{\eta}_1 + \ten{F}_y\times_{k\in[r]}\mat{\alpha}_k) \times_{l\in[k]}\mat{\Omega}_k^{1} \\


\invlink_2(\mat{\eta}_y) &\equiv \E[\vec(\ten{X})\t{\vec(\ten{X})} \mid Y = y] \\


&= \cov(\vec{X} \mid Y = y) + \vec(\ten{\mu}_y)\t{\vec(\ten{\mu}_y)} \\


&= \bigotimes_{k = r}^{1}\mat{\Omega}_k^{1} + \vec(\ten{\mu}_y)\t{\vec(\ten{\mu}_y)}


\end{align*}




For estimation purposes it's also of interest to express the logpartition function $b$ in terms of the natural parameters or the GLM parameters which has the form


\begin{displaymath}


b(\mat{\eta}_y) = \frac{1}{2}\t{\mat{\eta}_{y, 1}}(\unvec(2\mat{\eta}_{y, 2}))^{1}  \frac{1}{2}\log\unvec(2\mat{\eta}_{y, 2}).


\end{displaymath}






Denote the Residuals as


\begin{displaymath}


\ten{R}_i = \ten{X}_i  \ten{\mu}_{y_i}


\end{displaymath}


then with $\overline{\ten{R}} = \frac{1}{n}\sum_{i = 1}^n \ten{R}_i$ we get


\begin{align*}


\nabla_{\overline{\eta}_1} l &= \overline{\ten{R}}, \\


\nabla_{\mat{\alpha}_j} l &= \frac{1}{n}\ten{R}_{(j)}\t{(\ten{F}_y\times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}}, \\


\D l(\mat{\Omega}_j) &= \frac{1}{2}\t{\vec\Big(\frac{p}{p_j}\mat{\Omega}_j^{1}  (\ten{X} + \mu_y)_{(j)}\t{(\ten{R}\times_{k\in[r]\backslash j}\mat{\Omega}_k)_{(j)}}\Big)}\mat{D}_{p_j}\t{\mat{D}_{p_j}}


\end{align*}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsubsection{Initial Values}


First we set the gradient with respect to $\overline{\ten{\eta}}_1$ to zero


\begin{gather*}


0 \overset{!}{=} \nabla_{\overline{\ten{\eta}}_1}l = c_1\sum_{i = 1}^n (\ten{X}_i  \ten{\mu}_i) \\


\overline{\ten{X}} = (\overline{\ten{\eta}}_1 + \overline{\ten{F}_y}\times_{k\in[r]}\mat{\alpha}_{k})\times_{l\in[r]}\mat{\Omega}_{l}^{1} \\


\overline{\ten{X}}\times_{l\in[r]}\mat{\Omega}_{l} = \overline{\ten{\eta}}_1 + \overline{\ten{F}_y}\times_{k\in[r]}\mat{\alpha}_{k} \approx \overline{\ten{\eta}}_1 \\


\overline{\ten{\eta}}_1^{(0)} = \overline{\ten{X}}\times_{k\in[r]}\mat{\Omega}_{k}^{(0)}


\end{gather*}


where the approximation is due to the assumption that $\E \ten{F}_y = 0$. For the initial values of the scatter matrices $\mat{\Omega}_{l}$ we simply ignore the relation to the response and simply estimate them as the marginal scatter matrices. These initial estimates overemphasize the variability in the reduction subspace. Therefore, we first compute the unscaled mode covariance estimates


\begin{displaymath}


\widetilde{\mat{\Delta}}_j^{(0)} = \frac{p_j}{n p} (\ten{X}  \overline{\ten{X}})_{(j)}\t{(\ten{X}  \overline{\ten{X}})_{(j)}}.


\end{displaymath}


The next step is to scale them such that there Kronecker product has an appropriate trace


\begin{displaymath}


\mat{\Delta}_j^{(0)} = \left(\frac{\\ten{X}  \overline{\ten{X}}\_F^2}{n \prod_{k = 1}^r \tr(\widetilde{\mat{\Delta}}_j^{(0)})}\right)^{1 / r} \widetilde{\mat{\Delta}}_j^{(0)}.


\end{displaymath}


Finally, the covariances need to be inverted to give initial estimated of the scatter matrices


\begin{displaymath}


\mat{\Omega}_j^{(0)} = (\mat{\Delta}_j^{(0)})^{1}.


\end{displaymath}


The relay interesting part is to get initial estimates for the $\mat{\alpha}_j$ matrices. Setting the $\mat{\alpha}_j$ gradient to zero gives and substituting the initial estimates for $\overline{\ten{\eta}}_1$ and the $\mat{\Omega}_k$'s gives


\begin{gather*}


0 \overset{!}{=} \nabla_{\mat{\alpha}_j}l = c_1 \sum_{i = 1}^n \reshape{\mat{p}}(\mat{t}_1(\ten{X}_i)  \mat{g}_1(\mat{\eta}_{y_i}))_{(j)}\t{(\ten{F}_{y_i}\times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}} \\


(\ten{X}  \overline{\ten{X}})_{(j)}\t{(\ten{F}_y \times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}}


= \mat{\Omega}_j^{(0)}\mat{\alpha}_j(\ten{F}_y\times_{k\in[r]\backslash j}\mat{\Omega}_k^{(0)}\mat{\alpha}_k)_{(j)}\t{(\ten{F}_y \times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}}


\end{gather*}


Now letting $\mat{\Sigma}_k$ be the mode covariances of $\ten{F}_y$ and define $\ten{W}_y = \ten{F}_y\times_{k\in[r]}\mat{\Sigma}_k$ we get


\begin{gather*}


(\ten{X}  \overline{\ten{X}})_{(j)}\t{(\ten{F}_y \times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}}


= \mat{\Omega}_j^{(0)}\mat{\alpha}_j\mat{\Sigma}_j^{1/2}(\ten{W}_y\times_{k\in[r]\backslash j}\mat{\Omega}_k^{(0)}\mat{\alpha}_k\mat{\Sigma}_k^{1/2})_{(j)}\t{(\ten{W}_y \times_{k\in[r]\backslash j}\mat{\alpha}_k \mat{\Sigma}_{k}^{1/2})_{(j)}}\mat{\Sigma}_{j}^{1/2} \\


= \mat{\Omega}_j^{(0)}\mat{\alpha}_j\mat{\Sigma}_j^{1/2}(\ten{W}_y)_{(j)}\Big(\mat{I}_n\otimes\bigotimes_{\substack{k = r\\k\neq j}}^{1}\mat{\Sigma}_k^{1/2}\t{\mat{\alpha}_k}\mat{\Omega}_k^{(0)}\mat{\alpha}_k\mat{\Sigma}_{k}^{1/2}\Big)\t{(\ten{W}_y)_{(j)}}\mat{\Sigma}_{j}^{1/2}.


\end{gather*}


Now we let $\mat{\alpha}_j^{(0)}$ be such that $\mat{\Sigma}_k^{1/2}\t{(\mat{\alpha}^{(0)}_k)}\mat{\Omega}_k^{(0)}\mat{\alpha}^{(0)}_k\mat{\Sigma}_{k}^{1/2} = \mat{I}_{p_j}$, which leads by substitution to


\begin{displaymath}


(\ten{X}  \overline{\ten{X}})_{(j)}\t{(\ten{F}_y\times_{k\in[r]\backslash j}\mat{\alpha}^{(0)}_k)_{(j)}}


= \mat{\Omega}_j^{(0)}\mat{\alpha}^{(0)}_j\mat{\Sigma}_j^{1/2}(\ten{W}_y)_{(j)}\t{(\ten{W}_y)_{(j)}}\mat{\Sigma}_{j}^{1/2}


= \frac{p_j}{n p}\mat{\Omega}_j^{(0)}\mat{\alpha}^{(0)}_j\mat{\Sigma}_j


\end{displaymath}


\todo{Does this make sense?!?!?!}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Ising Model}


For the inverse regression $\ten{X}\mid Y = y$ the Ising model probability mass function with $p (p + 1) / 2$ parameters $\mat{\theta}_y$ is given by


\begin{align*}


P_{\mat{\theta}_y}(\ten{X}\mid Y = y)


&= p_0(\mat{\theta}_y)\exp(\t{\vech(\vec(\ten{X})\t{\vec(\ten{X})})}\mat{\theta}_y) \\


&= h(\ten{X})\exp(\t{\mat{{\eta}}(\mat{\theta}_y)}\mat{t}(\ten{X})  b(\mat{\theta}_y))


\end{align*}


where $h(\ten{X}) = 1$ and $b(\mat{\theta}_y) = \log p_0(\mat{\theta}(\mat{\eta}_y))$.


According to the GLM model we get


\begin{align*}


\mat{\eta}_{y,1} &\equiv c_1 (\overline{\ten{\eta}}_1 + \ten{F}_y\times_{k\in[r]}\mat{\alpha}_k), &


\mat{\eta}_{y,2} &\equiv c_2 \bigotimes_{k = r}^{1}\mat{\Omega}_k.


\end{align*}


which yields the following relation to the conditional Ising model parameters


\begin{displaymath}


\mat{\theta}_y = \mat{\theta}(\mat{\eta}_y) = \vech(\diag(\mat{\eta}_{y,1}) + (2_{p\times p}  \mat{I}_p) \odot \reshape{(p, p)}(\mat{\eta}_{y,2}))


\end{displaymath}


where the constants $c_1, c_2$ can be chosen arbitrary, as long as they are nonzero. The ``inverse'' link in then computed via the Ising model as the conditional expectation of all interactions


\begin{align*}


\invlink_2(\mat{\eta}_y) \equiv \E_{\mat{\theta}_y}[\vec(\ten{X})\t{\vec(\ten{X})}\mid Y = y]


\end{align*}


which incorporates the first moment. In other words $\invlink_1(\mat{\eta}_y) = \diag(\E_{\mat{\theta}_y}[\vec(\ten{X})\t{\vec(\ten{X})}\mid Y = y])$.






% The ``inverse'' link is given by


% \begin{align*}


% \invlink_1(\mat{\eta}_y) &\equiv \E_{\mat{\theta}(\mat{\eta}_y)}[\ten{X}  Y = y] \\


% \invlink_2(\mat{\eta}_y) &\equiv \E_{\mat{\theta}(\mat{\eta}_y)}[\vec(\ten{X})\t{\vec(\ten{X})}  Y = y]


% \end{align*}


% and note that $\diag(\E_{\mat{\theta}(\mat{\eta}_y)}[\vec(\ten{X})\t{\vec(\ten{X})}  Y = y]) \equiv \E_{\mat{\theta}(\mat{\eta}_y)}[\ten{X}  Y = y]$.


%


% The gradients of the loglikelihood are now given by


% \begin{align*}


% \nabla_{\overline{\ten{\eta}}_1} l


% &= \frac{1}{n}\sum_{i = 1}^n \ten{R}_i \\


% \nabla_{\mat{\alpha}_j} l


% &= \frac{1}{n}\ten{R}_{(j)}\t{(\ten{F}_y\times_{k\in[r]\backslash j}\mat{\alpha}_j)_{(j)}} \\


% \vec(\nabla_{\mat{\Omega}_j} l)


% &= \t{\vec( (\reshape{(\mat{p}, \mat{p})}(\overline{\mat{t}_2(\ten{X}_i)}  \E[\mat{t}_2(\ten{X})\mid Y = y_i]))_{(j, r + j)} \vec\bigotimes_{\substack{k = r\\k\neq j}}^{1}\mat{\Omega}_j)}\mat{D}_{p_j}\t{\mat{D}_{p_j}}


% \end{align*}


% using the notation $\overline{\mat{t}_2(\ten{X})} = \frac{1}{n}\sum_{i = 1}^n \mat{t}_2(\ten{X}_i) = \frac{1}{n}\sum_{i = 1}^n \vec(\ten{X}_i)\otimes \vec(\ten{X}_i)$.




\printbibliography[heading=bibintoc,title={References}]


\appendix


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Vectorization and Matricization}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{displaymath}


\vec(\ten{A}\times_{k\in[r]}\mat{B}_k) = \Big(\bigotimes_{k = r}^1 \mat{B}_k\Big)\vec\ten{A}


\end{displaymath}


\begin{displaymath}


(\ten{A}\times_{k\in[r]}\mat{B}_k)_{(j)} = \mat{B}_j\ten{A}_{(j)}\bigotimes_{\substack{k = r\\k\neq j}}^1\t{\mat{B}_k}


\end{displaymath}


of which a special case is $(\ten{A}\times_{j}\mat{B}_j)_{(j)} = \mat{B}_j\ten{A}_{(j)}$.






Let $\ten{A}$ be a $n\times p_1\times ... \times p_r\times q_1\times ... \times q_r$ tensor and $\mat{B}_k$ be $p_k\times q_k$ matrices, then


\begin{displaymath}


\ten{A}_{(1)} \vec{\bigotimes_{k = r}^{1}\mat{B}_k}


=


\Big(\ten{R}(\ten{A})\times_{\substack{k + 1\\k\in[r]}}\t{\vec(\mat{B}_k)}\Big)_{(1)}


\end{displaymath}


where $\ten{R}$ is a permutation of the axis and reshaping of the tensor $\ten{A}$. This axis permutation converts $n\times p_1\times ... \times p_r\times q_1\times ... \times q_r$ to $n\times p_1\times q_1 \times ... \times p_r\times q_r$ and the reshaping vectorizes the axis pairs $p_k\times q_k$ leading to a tensor $\ten{R}(\ten{A})$ of dimensions $n\times p_1 q_1\times ...\times p_r q_r$.




An alternative way to write this is for each of the $i\in[n]$ vector components is


\begin{displaymath}


\Big(\ten{A}_{(1)}\vec{\bigotimes_{k = r}^{1}\mat{B}_k}\Big)_{i}


= \sum_{J\in[(\mat{p}, \mat{q})]}


\ten{A}_{i, J}\prod_{k = 1}^r (B_k)_{J_k, J_{k + r}}


\end{displaymath}


using the notation $J\in[(\mat{p}, \mat{q})] = [p_1]\times ... \times [p_r]\times [q_1]\times ... \times [q_r]$.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Pattern Matrices}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




The \emph{duplication matrix} $\mat{D}_p$ of dimensions $p^2\times p(p + 1) / 2$ is defined implicitly such that for any symmetric $p\times p$ matrix $\mat{A}$ holds


\begin{displaymath}


\mat{D}_p\vech\mat{A} = \vec{\mat{A}}.


\end{displaymath}


Let $\mat{A}$ by a $p\times q$ matrix, then we denote the \emph{commutation matrix} $\mat{K}_{p,q}$ as the $p q\times p q$ matrix satisfying


\begin{displaymath}


\mat{K}_{p,q}\vec\mat{A} = \vec{\t{\mat{A}}}.


\end{displaymath}


The identity giving the commutation matrix its name is


\begin{displaymath}


\mat{A}\otimes\mat{B} = \mat{K}_{a_1,b_1}(\mat{B}\otimes\mat{A})\t{\mat{K}_{a_2,b_2}}.


\end{displaymath}


For a generalization of the commutation matrix let $\ten{A}$ be a $p_1\times ...\times p_r$ tensor of order $r$. Then the \emph{generalized commutation matrix} $\mat{K}_{(p_1, ..., p_r),(j)}$ is implicitly defined such that


\begin{displaymath}


\mat{K}_{(p_1, ..., p_r),(j)}\vec{\ten{A}} = \vec{\ten{A}_{(j)}}


\end{displaymath}


for every $j \in[r]$ mode. This is a direct generalization of the commutation matrix with the special case $\mat{K}_{(p,q),(2)} = \mat{K}_{p,q}$ and the trivial case $\mat{K}_{(p_1, ..., p_r),(1)} = \mat{I}_{p}$ for $p = \prod_{j=1}^r p_j$. Furthermore, with a dimension vector $\mat{p} = (p_1, ..., p_r)$ its convenient to write $\mat{K}_{(p_1, ..., p_r),(j)}$ as $\mat{K}_{\mat{p},(j)}$. Its relation to the classic Commutation matrix is given by


\begin{displaymath}


\mat{K}_{\mat{p}, (j)} = \mat{I}_{\overline{p}_j} \otimes \mat{K}_{\underline{p}_j, p_j}


\end{displaymath}


where $\overline{p}_j = \prod_{k = j + 1}^r p_k$ and $\underline{p}_j = \prod_{k = 1}^{j  1}p_k$ with an empty product set to $1$.


The generalized commutation matrix gives leads to a generalization of the Kronecker product commutation identity


\begin{displaymath}


\bigotimes_{\substack{k = r\\k\neq j}}^{1}\mat{A}_k\otimes \mat{A}_j = \mat{K}_{\mat{p}, (j)}\Big(\bigotimes_{k = r}^1 \mat{A}_k\Big)\t{\mat{K}_{\mat{q}, (j)}}


\end{displaymath}


for arbitrary matrices $\mat{A}_k$ of dimensions $p_k\times q_k$, $k \in[r]$ which are collected in the dimension vectors $\mat{p} = (p_1, ..., p_r)$ and $\mat{q} = (q_1, ..., q_r)$. Next the \emph{symmetrizer} $\mat{N}_p$ is a $p^2\times p^2$ matrix such that for any $p\times p$ matrix $\mat{A}$


\begin{displaymath}


\mat{N}_p \vec{\mat{A}} = \frac{1}{2}(\vec{\mat{A}} + \vec{\t{\mat{A}}}).


\end{displaymath}


Another matrix which might come in handy is the \emph{selection matrix} $\mat{S}_p$ of dimensions $p^2\times p$ which selects the diagonal elements of a $p\times p$ matrix $\mat{A}$ from its vectorization


\begin{displaymath}


\mat{S}_p\vec{\mat{A}} = \diag{\mat{A}}


\end{displaymath}


where $\diag{\mat{A}}$ denotes the vector of diagonal elements of $\mat{A}$.




For two matrices $\mat A$ of dimensions $a_1\times a_2$ and $\mat B$ of dimensions $b_1\times b_2$ holds


\begin{equation}\label{eq:vecKron}


\vec(\mat A\otimes\mat B) = (\mat{I}_{a_2}\otimes\mat{K}_{b_2,a_1}\otimes\mat{I}_{b_1})(\vec\mat A\otimes\vec\mat B).


\end{equation}




\begin{align*}


\pinv{\mat{D}_p} &= (\t{\mat{D}_p}\mat{D}_p)^{1}\t{\mat{D}_p} \\


\pinv{\mat{D}_p}\mat{D}_p &= \mat{I}_{p(p+1)/2} \\


\mat{D}_p\pinv{\mat{D}_p} &= \mat{N}_{p} \\


\t{\mat{K}_{p,q}} &= \mat{K}_{p,q}^{1} = \mat{K}_{q,p} \\


\t{\mat{K}_{\mat{p},(j)}} &= \mat{K}_{\mat{p},(j)}^{1}


\end{align*}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Matrix Calculus}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\begin{example}


We want to find the derivative with respect to any of the $r$ symmetric $p_j\times p_j$ matrices $\mat{\Omega}_j$ where $j = 1, ..., r$ of the Kronecker product


\begin{displaymath}


\mat{F} = \bigotimes_{k = r}^1 \mat{\Omega}_k.


\end{displaymath}


Therefore, denote


\begin{align*}


p &= \prod_{k = 1}^r p_k, & \overline{p}_j &= \prod_{k = j + 1}^r p_k, & \underline{p}_j &= \prod_{k = 1}^{j  1} p_k, \\


& & \overline{\mat{\Omega}}_j &= \bigotimes_{k = r}^{j+1}\mat{\Omega}_k, & \underline{\mat{\Omega}}_j &= \bigotimes_{k = j  1}^{1}\mat{\Omega}_k


\end{align*}


which slightly simplifies the following. With this notation we have $p = \overline{p}_jp_j\underline{p}_j$ for any of the $j = 1, ..., r$. Furthermore, the matrices $\overline{\mat{\Omega}}_j$ and $\underline{\mat{\Omega}}_j$ are of dimensions $\overline{p}_j\times \overline{p}_j$ and $\underline{p}_j\times \underline{p}_j$, respectively. We start with the differential


\begin{align*}


\d\mat{F} &= \d\bigotimes_{k = r}^1 \mat{\Omega}_k


= \sum_{j = 1}^r \bigotimes_{k = r}^{j+1}\mat{\Omega}_k\otimes\d\mat{\Omega}_j\otimes\bigotimes_{k = j  1}^{1}\mat{\Omega}_k


= \sum_{j = 1}^r \overline{\mat{\Omega}}_j\otimes\d\mat{\Omega}_j\otimes\underline{\mat{\Omega}}_j \\


&= \sum_{j = 1}^r \mat{K}_{\overline{p}_jp_j,\underline{p}_j}(\underline{\mat{\Omega}}_j\otimes\overline{\mat{\Omega}}_j\otimes\d\mat{\Omega}_j)\mat{K}_{\underline{p}_j,\overline{p}_jp_j}


\end{align*}


By vectorizing this transforms to


\begin{align*}


\d\vec\mat{F} &= \sum_{j = 1}^r (\mat{K}_{\overline{p}_jp_j,\underline{p}_j}\otimes\mat{K}_{\overline{p}_jp_j,\underline{p}_j})\vec(\underline{\mat{\Omega}}_j\otimes\overline{\mat{\Omega}}_j\otimes\d\mat{\Omega}_j) \\


&= \sum_{j = 1}^r (\mat{K}_{\overline{p}_jp_j,\underline{p}_j}\otimes\mat{K}_{\overline{p}_jp_j,\underline{p}_j})(\mat{I}_{\overline{p}_j\underline{p}_j}\otimes\mat{K}_{p_j,\overline{p}_j\underline{p}_j}\otimes\mat{I}_{p_j})(\vec(\underline{\mat{\Omega}}_j\otimes\overline{\mat{\Omega}}_j)\otimes\d\vec\mat{\Omega}_j) \\


&= \sum_{j = 1}^r (\mat{K}_{\overline{p}_jp_j,\underline{p}_j}\otimes\mat{K}_{\overline{p}_jp_j,\underline{p}_j})(\mat{I}_{\overline{p}_j\underline{p}_j}\otimes\mat{K}_{p_j,\overline{p}_j\underline{p}_j}\otimes\mat{I}_{p_j})(\vec(\underline{\mat{\Omega}}_j\otimes\overline{\mat{\Omega}}_j)\otimes\mat{I}_{p_j^2})\,\d\vec\mat{\Omega}_j \\


\end{align*}


leading to


\begin{displaymath}


\D\mat{F}(\mat{\Omega}_j) = (\mat{K}_{\overline{p}_jp_j,\underline{p}_j}\otimes\mat{K}_{\overline{p}_jp_j,\underline{p}_j})(\mat{I}_{\overline{p}_j\underline{p}_j}\otimes\mat{K}_{p_j,\overline{p}_j\underline{p}_j}\otimes\mat{I}_{p_j})(\vec(\underline{\mat{\Omega}}_j\otimes\overline{\mat{\Omega}}_j)\otimes\mat{I}_{p_j^2})


\end{displaymath}


for each $j = 1, ..., r$. Note that the $p^2\times p^2$ matrices


\begin{displaymath}


\mat{P}_j = (\mat{K}_{\overline{p}_jp_j,\underline{p}_j}\otimes\mat{K}_{\overline{p}_jp_j,\underline{p}_j})(\mat{I}_{\overline{p}_j\underline{p}_j}\otimes\mat{K}_{p_j,\overline{p}_j\underline{p}_j}\otimes\mat{I}_{p_j})


\end{displaymath}


are permutations.


\end{example}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Stuff}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Let $X, Y$ be $p, q$ dimensional random variables, respectively. Furthermore, let $\E X = \mu_X$, $\E Y = \mu_Y$ as well as $\cov(X) = \mat{\Sigma}_X$ and $\cov(Y) = \mat{\Sigma}_Y$. Then define the standardized random variables $Z_X = \mat{\Sigma}_X^{1/2}(X  \mu_X)$ and $Z_Y = \mat{\Sigma}_Y^{1/2}(Y  \mu_Y)$. For the standardized variables holds $\E Z_X = 0_p$, $\E_Y = 0_q$ and for the covariances we get $\cov(Z_X) = \mat{I}_p$ as well as $\cov(Z_Y) = \mat{I}_q$. Now we take a look at the crosscovariance between $X$ and $Y$


\begin{displaymath}


\cov(X, Y)


= \cov(X  \mu_X, Z  \mu_Z)


= \cov(\mat{\Sigma}_X^{1/2} Z_X, \mat{\Sigma}_Y^{1/2} Z_Y)


= \mat{\Sigma}_X^{1/2}\cov(Z_X, Z_Y)\mat{\Sigma}_Y^{1/2}.


\end{displaymath}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Proofs}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{proof}[Proof of Theorem~\ref{thm:sdr}]


abc


\end{proof}




\end{document}


@ 0,0 +1,656 @@


\documentclass[a4paper, 10pt]{article}




\usepackage[utf8]{inputenc}


\usepackage[T1]{fontenc}


\usepackage{fullpage}


\usepackage{amsmath, amssymb, amstext, amsthm}


\usepackage{bm} % \boldsymbol and italic corrections, ...


\usepackage[pdftex]{hyperref}


\usepackage{makeidx} % Index (Symbols, Names, ...)


\usepackage{xcolor, graphicx} % colors and including images


\usepackage{tikz}


\usetikzlibrary{calc}


\usepackage[


% backend=bibtex,


style=authoryearcomp


]{biblatex}


\usepackage{algorithm, algpseudocode} % Pseudo Codes / Algorithms




% Document meta into


\title{Bernoulli}


\author{Daniel Kapla}


\date{\today}


% Set PDF title, author and creator.


\AtBeginDocument{


\hypersetup{


pdftitle = {Bernoulli},


pdfauthor = {Daniel Kapla},


pdfcreator = {\pdftexbanner}


}


}




\makeindex




% Bibliography resource(s)


\addbibresource{main.bib}




% Setup environments


% Theorem, Lemma


\theoremstyle{plain}


\newtheorem{theorem}{Theorem}


\newtheorem{lemma}{Lemma}


\newtheorem{example}{Example}


% Definition


\theoremstyle{definition}


\newtheorem{defn}{Definition}


% Remark


\theoremstyle{remark}


\newtheorem{remark}{Remark}




% Define math macros


\newcommand{\mat}[1]{\boldsymbol{#1}}


\newcommand{\ten}[1]{\mathcal{#1}}


\renewcommand{\vec}{\operatorname{vec}}


\newcommand{\dist}{\operatorname{dist}}


\newcommand{\rank}{\operatorname{rank}}


\DeclareMathOperator{\kron}{\otimes} % Kronecker Product


\DeclareMathOperator{\hada}{\odot} % Hadamard Product


\newcommand{\ttm}[1][n]{\times_{#1}} % nmode product (Tensor Times Matrix)


\DeclareMathOperator{\df}{df}


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\DeclareMathOperator{\var}{Var}


\DeclareMathOperator{\cov}{Cov}


\DeclareMathOperator{\Span}{Span}


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% \DeclareMathOperator{\independent}{{\bot\!\!\!\bot}}


\DeclareMathOperator*{\argmin}{{arg\,min}}


\DeclareMathOperator*{\argmax}{{arg\,max}}


\newcommand{\D}{\textnormal{D}} % derivative


\renewcommand{\d}{\textnormal{d}} % differential


\renewcommand{\t}[1]{{#1^{\prime}}} % matrix transpose


\newcommand{\pinv}[1]{{#1^{\dagger}}} % `MoorePenrose pseudoinverse`


\renewcommand{\vec}{\operatorname{vec}}


\newcommand{\vech}{\operatorname{vech}}


\newcommand{\logical}[1]{{[\![#1]\!]}}




\begin{document}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Bivariate Bernoulli Distribution}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




A random 2vector $X\in\{0, 1\}^2$ follows a \emph{Bivariate Bernoulli} distribution if its pmf is


\begin{displaymath}


P(X = (x_1, x_2)) = p_{00}^{(1x_1)(1x_2)}p_{01}^{(1x_1)x_2}p_{10}^{x_1(1x_2)}p_{11}^{x_1x_2}


\end{displaymath}


where $p_{ab} = P(X = (a, b))$ for $a, b\in\{0, 1\}$. An alternative formulation, in terms of logodds, follows immediately as


\begin{displaymath}


P(X = (x_1, x_2)) = p_{00}\exp\Big(x_1\log\frac{p_{10}}{p_{00}} + x_2\log\frac{p_{01}}{p_{00}} + x_1x_2\log\frac{p_{00}p_{11}}{p_{01}p_{10}}\Big).


\end{displaymath}


Collecting the logodds in a parameter vector $\mat{\theta} = \t{(\theta_{01}, \theta_{10}, \theta_{11})}$ where


\begin{align*}


\theta_{01} &= \log\frac{p_{01}}{p_{00}}, \\


\theta_{10} &= \log\frac{p_{10}}{p_{00}}, \\


\theta_{11} &= \log\frac{p_{00}p_{11}}{p_{01}p_{10}}


\end{align*}


the pmf can be written more compact as


\begin{displaymath}


P(X = (x_1, x_2)) = P(X = \mat{x}) = p_{00}\exp(\t{\mat{\theta}}\vech(\mat{x}\t{\mat{x}}))


= p_{00}\exp(\t{\mat{x}}\mat{\Theta}\mat{x})


\end{displaymath}


with the parameter matrix $\mat{\Theta}$ defined as


\begin{displaymath}


\mat{\Theta} = \begin{pmatrix}


\theta_{01} & \tfrac{1}{2}\theta_{11} \\


\tfrac{1}{2}\theta_{11} & \theta_{10}


\end{pmatrix} = \begin{pmatrix}


\log\frac{p_{01}}{p_{00}} & \tfrac{1}{2}\log\frac{p_{00}p_{11}}{p_{01}p_{10}} \\


\tfrac{1}{2}\log\frac{p_{00}p_{11}}{p_{01}p_{10}} & \log\frac{p_{10}}{p_{00}}


\end{pmatrix}.


\end{displaymath}


The marginal distribution of $X_1$ and $X_2$ are given by


\begin{align*}


P(X_1 = x_1) &= P(X = (x_1, 0)) + P(X = (x_1, 1)) \\


&= p_{00}^{1x_1}p_{10}^{x_1} + p_{01}^{1x_1}p_{11}^{x_1} \\


&= \begin{cases}


p_{00} + p_{01}, & x_1 = 0 \\


p_{01} + p_{11}, & x_1 = 1


\end{cases} \\


&= (p_{00} + p_{01})^{1x_1}(p_{01} + p_{11})^{x_1}. \\


P(X_2 = x_2) &= (p_{00} + p_{10})^{1x_2}(p_{10} + p_{11})^{x_2}.


\end{align*}


Furthermore, the conditional distributions are


\begin{align*}


P(X_1 = x_1X_2 = x_2) = \frac{P(X = (x_1, x_2))}{P(X_2 = x_2)}


\propto \big(p_{00}^{1x_2}p_{01}^{x_2}\big)^{1x_1}\big(p_{10}^{1x_2}p_{11}^{x_2}\big)^{x_1}, \\


P(X_2 = x_2X_1 = x_1) = \frac{P(X = (x_1, x_2))}{P(X_1 = x_1)}


\propto \big(p_{00}^{1x_1}p_{10}^{x_1}\big)^{1x_2}\big(p_{01}^{1x_1}p_{11}^{x_1}\big)^{x_2}.


\end{align*}


Note that both the marginal and the conditional are again Bernoulli distributed. Its also of interest to look at the covariance between the components of $X$ which are given by


\begin{displaymath}


\cov(X_1, X_2) = \E[(X_1  \E X_1)(X_2  \E X_2)] = p_{00}p_{11}  p_{01}p_{10}


\end{displaymath}


which follows by direct computation.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Multivariate Bernoulli Distribution}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




This is a direct generalization of the Bivariate Bernoulli Distribution. Before we start a few notes on notation. Let $a, b$ be binary vectors, then $\logical{a = b} = 1$ if and only if $\forall i : a_i = b_i$ and zero otherwise. With that, let $Y\in\{0, 1\}^q$ be a $q$dimensional \emph{Multivariate Bernoulli} random variable with pdf


\begin{equation}\label{eq:mvb_pmf}


P(Y = y) = \prod_{a\in\{0, 1\}^q} p_a^{\logical{y = a}} = p_y.


\end{equation}


The parameters are $2^q$ parameters $p_a$ which are indexed by the event $a\in\{0, 1\}^q$. The ``indexing'' is done by identifying an event $a\in\{0, 1\}^q$ with the corresponding binary number $m$ the event represents. In more detail we equate an event $a\in\{0, 1\}^q$ with a number $m\in[0; 2^q  1]$ as


\begin{equation}\label{eq:mvb_numbering}


m = m(a) = \sum_{i = 1}^q 2^{q  i}a_i


\end{equation}


which is a onetoone relation. For example, for $q = 3$ all events are numbered as in Table~\ref{tab:eventtonumber}.


\begin{table}[!ht]


\centering


\begin{minipage}{0.8\textwidth}


\centering


\begin{tabular}{cc}


Event $a$ & Number $m(a)$ \\ \hline


(0, 0, 0) & 0 \\


(0, 0, 1) & 1 \\


(0, 1, 0) & 2 \\


(0, 1, 1) & 3 \\


(1, 0, 0) & 4 \\


(1, 0, 1) & 5 \\


(1, 1, 0) & 6 \\


(1, 1, 1) & 7


\end{tabular}


\caption{\label{tab:eventtonumber}\small Event numbering relation for $q = 3$. The events $a$ are all the possible elements of $\{0, 1\}^3$ and the numbers $m$ range from $0$ to $2^3  1 = 7$.}


\end{minipage}


\end{table}




\subsection{Exponential Family and Natural Parameters}


The Multivariate Bernoulli is a member of the exponential family. This can be seen by rewriting the pmf \eqref{eq:mvb_pmf} in terms of an exponential family. First, we take a look at $\logical{y = a}$ for two binary vectors $y, a$ which can be written as


\begin{align*}


\logical{y = a}


&= \prod_{i = 1}^q (y_i a 