add: kpir.ls as initial value estimate (alternative to VLP)

This commit is contained in:
Daniel Kapla 2022-05-18 18:33:28 +02:00
parent 49bf4bdf20
commit a2963024ef
8 changed files with 315 additions and 162 deletions

9
.gitignore vendored
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@ -81,8 +81,12 @@ vignettes/*.pdf
*.RData *.RData
*.Rdata *.Rdata
# R Profiling
*.Rprof
# VSCode configuration # VSCode configuration
.vscode/ .vscode/
.lintr
## Archives, compressed files/folders ## Archives, compressed files/folders
# Output files from R CMD build # Output files from R CMD build
@ -104,3 +108,8 @@ wip/
mlda_analysis/ mlda_analysis/
References/ References/
# Images (except images used in LaTeX)
*.png
*.svg
!**/LaTeX/*.png

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@ -49,6 +49,7 @@
\newcommand{\mat}[1]{\boldsymbol{#1}} \newcommand{\mat}[1]{\boldsymbol{#1}}
\newcommand{\ten}[1]{\mathcal{#1}} \newcommand{\ten}[1]{\mathcal{#1}}
\renewcommand{\vec}{\operatorname{vec}} \renewcommand{\vec}{\operatorname{vec}}
\newcommand{\dist}{\operatorname{dist}}
\DeclareMathOperator{\kron}{\otimes} % Kronecker Product \DeclareMathOperator{\kron}{\otimes} % Kronecker Product
\DeclareMathOperator{\hada}{\odot} % Hadamard Product \DeclareMathOperator{\hada}{\odot} % Hadamard Product
\newcommand{\ttm}[1][n]{\times_{#1}} % n-mode product (Tensor Times Matrix) \newcommand{\ttm}[1][n]{\times_{#1}} % n-mode product (Tensor Times Matrix)
@ -62,8 +63,8 @@
\DeclareMathOperator*{\argmax}{{arg\,max}} \DeclareMathOperator*{\argmax}{{arg\,max}}
\newcommand{\D}{\textnormal{D}} \newcommand{\D}{\textnormal{D}}
\renewcommand{\d}{\textnormal{d}} \renewcommand{\d}{\textnormal{d}}
\renewcommand{\t}[1]{{{#1}'}} \renewcommand{\t}[1]{{#1^{\prime}}}
\newcommand{\pinv}[1]{{{#1}^{\dagger}}} % `Moore-Penrose pseudoinverse` \newcommand{\pinv}[1]{{#1^{\dagger}}} % `Moore-Penrose pseudoinverse`
\newcommand{\todo}[1]{{\color{red}TODO: #1}} \newcommand{\todo}[1]{{\color{red}TODO: #1}}
% \DeclareFontFamily{U}{mathx}{\hyphenchar\font45} % \DeclareFontFamily{U}{mathx}{\hyphenchar\font45}
@ -117,17 +118,15 @@ Let $\ten{X}$ be a multi-dimensional array random variable of order (rank) $r$ w
\end{displaymath} \end{displaymath}
Its density is given by Its density is given by
\begin{displaymath} \begin{displaymath}
f(\ten{X}) = \Big( \prod_{i = 1}^r \sqrt{(2\pi)^{p_i}|\mat{\Delta}_i|^{q_i}} \Big)^{-1} f(\ten{X}) = \Big( \prod_{i = 1}^r \sqrt{(2\pi)^{p_i}|\mat{\Delta}_i|^{p_{-i}}} \Big)^{-1}
\exp\!\left( -\frac{1}{2}\langle \ten{X} - \mu, (\ten{X} - \mu)\times\{\mat{\Delta}_1^{-1}, ..., \mat{\Delta}_r^{-1}\} \rangle \right) \exp\!\left( -\frac{1}{2}\langle \ten{X} - \mu, (\ten{X} - \mu)\times\{\mat{\Delta}_1^{-1}, ..., \mat{\Delta}_r^{-1}\} \rangle \right)
\end{displaymath} \end{displaymath}
where $q_i = \prod_{j \neq i}p_j$. This is equivalent to the vectorized $\vec\ten{X}$ following a Multi-Variate Normal distribution where $p_{\lnot i} = \prod_{j \neq i}p_j$. This is equivalent to the vectorized $\vec\ten{X}$ following a Multi-Variate Normal distribution
\begin{displaymath} \begin{displaymath}
\vec{\ten{X}}\sim\mathcal{N}_{p}(\vec{\mu}, \mat{\Delta}_r\otimes...\otimes\mat{\Delta}_1) \vec{\ten{X}}\sim\mathcal{N}_{p}(\vec{\mu}, \mat{\Delta}_r\otimes...\otimes\mat{\Delta}_1)
\end{displaymath} \end{displaymath}
with $p = \prod_{i = 1}^r p_i$. with $p = \prod_{i = 1}^r p_i$.
\begin{theorem}[Tensor Normal to Multi-Variate Normal equivalence] \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 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
\begin{displaymath} \begin{displaymath}
@ -148,12 +147,12 @@ with $p = \prod_{i = 1}^r p_i$.
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 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
\begin{displaymath} \begin{displaymath}
|\mat{\Delta}_r\otimes...\otimes\mat{\Delta}_1| |\mat{\Delta}_r\otimes...\otimes\mat{\Delta}_1|
= |\mat{\Delta}_r\otimes...\otimes\mat{\Delta}_2|^{p_1}|\mat{\Delta}_1|^{q_1} = |\mat{\Delta}_r\otimes...\otimes\mat{\Delta}_2|^{p_1}|\mat{\Delta}_1|^{p_{\lnot 1}}
\end{displaymath} \end{displaymath}
where $q_i = \prod_{j \neq i}p_j$. By induction over $r$ the relation where $p_{\lnot i} = \prod_{j \neq i}p_j$. By induction over $r$ the relation
\begin{displaymath} \begin{displaymath}
|\mat{\Delta}_r\otimes...\otimes\mat{\Delta}_1| |\mat{\Delta}_r\otimes...\otimes\mat{\Delta}_1|
= \prod_{i = 1}^r |\mat{\Delta}_i|^{q_i} = \prod_{i = 1}^r |\mat{\Delta}_i|^{p_{\lnot i}}
\end{displaymath} \end{displaymath}
holds for arbitrary order $r$. Substituting into the Tensor Normal density leads to holds for arbitrary order $r$. Substituting into the Tensor Normal density leads to
\begin{align*} \begin{align*}
@ -163,6 +162,14 @@ with $p = \prod_{i = 1}^r p_i$.
which is the Multi-Variate Normal density of the $p$ dimensional vector $\vec\ten{X}$. which is the Multi-Variate Normal density of the $p$ dimensional vector $\vec\ten{X}$.
\end{proof} \end{proof}
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
\begin{displaymath}
\ten{Z}\sim\mathcal{TN}(0, \mat{I}_{p_1}, ..., \mat{I}_{p_r})
\quad\Rightarrow\quad
\ten{X} = \ten{Z}\times\{\mat{\Delta}_1^{1/2}, ..., \mat{\Delta}_r^{1/2}\} + \mu\sim\mathcal{TN}(\mu, \mat{\Delta}_1, ..., \mat{\Delta}_r).
\end{displaymath}
where the sampling from the standard Multi-Array Normal is done by sampling all of the elements of $\ten{Z}$ from a standard Normal.
\section{Introduction} \section{Introduction}
We assume the model We assume the model
@ -385,7 +392,7 @@ The unbiasednes comes directly from the following short computation;
= \frac{1}{n}\sum_{i = 1}^n \sum_{l = 1}^p (\mat{\Delta}_{2})_{l,l}(\mat{\Delta}_{1})_{j,k} = \frac{1}{n}\sum_{i = 1}^n \sum_{l = 1}^p (\mat{\Delta}_{2})_{l,l}(\mat{\Delta}_{1})_{j,k}
= (\mat\Delta_1\tr(\mat\Delta_2))_{j,k}. = (\mat\Delta_1\tr(\mat\Delta_2))_{j,k}.
\end{displaymath} \end{displaymath}
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 unbiased by 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
\begin{displaymath} \begin{displaymath}
\tilde{s} = \frac{1}{n}\sum_{i = 1}^n \|\mat{R}_i\|_F^2 \tilde{s} = \frac{1}{n}\sum_{i = 1}^n \|\mat{R}_i\|_F^2
\end{displaymath} \end{displaymath}
@ -481,7 +488,7 @@ Putting it all together
&\hspace{3em}\hspace{3em} \hspace{4.7em} - \tilde{s}(\widetilde{\mat{\Delta}}_1^{-1}\mat{S}_1\widetilde{\mat{\Delta}}_1^{-1}\t{\mat{R}_i} + \t{\mat{R}_i}\widetilde{\mat{\Delta}}_2^{-1}\mat{S}_2\widetilde{\mat{\Delta}}_2^{-1} - \widetilde{\mat{\Delta}}_1^{-1} \t{\mat{R}_i} \widetilde{\mat{\Delta}}_2^{-1}) &\hspace{3em}\hspace{3em} \hspace{4.7em} - \tilde{s}(\widetilde{\mat{\Delta}}_1^{-1}\mat{S}_1\widetilde{\mat{\Delta}}_1^{-1}\t{\mat{R}_i} + \t{\mat{R}_i}\widetilde{\mat{\Delta}}_2^{-1}\mat{S}_2\widetilde{\mat{\Delta}}_2^{-1} - \widetilde{\mat{\Delta}}_1^{-1} \t{\mat{R}_i} \widetilde{\mat{\Delta}}_2^{-1})
\Big)\d\mat{\beta}\bigg). \Big)\d\mat{\beta}\bigg).
\end{align*} \end{align*}
Observe that the bracketed expressions before $\d\mat{\alpha}$ and $\d\mat{\beta}$ are transposed of each other. Lets denote the expression for $\d\mat{\alpha}$ as $\mat{G}_i$ which has the form 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} \begin{displaymath}
\mat{G}_i \mat{G}_i
= (\tr(\widetilde{\mat{\Delta}}_1^{-1}\mat{S}_1) - p q \tilde{s}^{-1})\mat{R}_i = (\tr(\widetilde{\mat{\Delta}}_1^{-1}\mat{S}_1) - p q \tilde{s}^{-1})\mat{R}_i
@ -521,7 +528,7 @@ Let $\ten{X}, \ten{F}$ be order (rank) $r$ tensors of dimensions $p_1\times ...
\begin{displaymath} \begin{displaymath}
\ten{X}_i = \ten{\mu} + \ten{F}_i\times\{ \mat{\alpha}_1, ..., \mat{\alpha}_r \} + \ten{\epsilon}_i \ten{X}_i = \ten{\mu} + \ten{F}_i\times\{ \mat{\alpha}_1, ..., \mat{\alpha}_r \} + \ten{\epsilon}_i
\end{displaymath} \end{displaymath}
where we assume $\ten{\epsilon}_i$ to be i.i.d. mean zero tensor normal distributed $\ten{\epsilon}\sim\mathcal{TM}(0, \mat{\Delta}_1, ..., \mat{\Delta}_r)$ for $\mat{\Delta}_j\in\mathcal{S}^{p_j}_{++}$, $j = 1, ..., r$. Given $i = 1, ..., n$ observations the collected model containing all observations where we assume $\ten{\epsilon}_i$ to be i.i.d. mean zero tensor normal distributed $\ten{\epsilon}\sim\mathcal{TN}(0, \mat{\Delta}_1, ..., \mat{\Delta}_r)$ for $\mat{\Delta}_j\in\mathcal{S}^{p_j}_{++}$, $j = 1, ..., r$. Given $i = 1, ..., n$ observations the collected model containing all observations
\begin{displaymath} \begin{displaymath}
\ten{X} = \ten{\mu} + \ten{F}\times\{ \mat{\alpha}_1, ..., \mat{\alpha}_r, \mat{I}_n \} + \ten{\epsilon} \ten{X} = \ten{\mu} + \ten{F}\times\{ \mat{\alpha}_1, ..., \mat{\alpha}_r, \mat{I}_n \} + \ten{\epsilon}
\end{displaymath} \end{displaymath}
@ -529,48 +536,87 @@ which is almost identical as the observations $\ten{X}_i, \ten{F}_i$ are stacked
In the following we assume w.l.o.g that $\ten{\mu} = 0$, as if this is not true we simply replace $\ten{X}_i$ with $\ten{X}_i - \ten{\mu}$ for $i = 1, ..., n$ before collecting all the observations in the response tensor $\ten{X}$. In the following we assume w.l.o.g that $\ten{\mu} = 0$, as if this is not true we simply replace $\ten{X}_i$ with $\ten{X}_i - \ten{\mu}$ for $i = 1, ..., n$ before collecting all the observations in the response tensor $\ten{X}$.
The goal here is to find reasonable estimates for $\mat{\alpha}_i$, $i = 1, ..., n$ for the mean model The goal here is to find reasonable estimates for $\mat{\alpha}_j$, $j = 1, ..., n$ for the mean model
\begin{displaymath} \begin{displaymath}
\E \ten{X}|\ten{F}, \mat{\alpha}_1, ..., \mat{\alpha}_r = \ten{F}\times\{\mat{\alpha}_1, ..., \mat{\alpha}_r, \mat{I}_n\} \E \ten{X}|\ten{F}, \mat{\alpha}_1, ..., \mat{\alpha}_r = \ten{F}\times\{\mat{\alpha}_1, ..., \mat{\alpha}_r, \mat{I}_n\}
= \ten{F}\times_{i\in[r]}\mat{\alpha}_i. = \ten{F}\times_{j\in[r]}\mat{\alpha}_j.
\end{displaymath} \end{displaymath}
Under the mean model we have using the general mode product relation $(\ten{A}\times_j\mat{B})_{(j)} = \mat{B}\ten{A}_{(j)}$ we get Under the mean model we have using the general mode product relation $(\ten{A}\times_j\mat{B})_{(j)} = \mat{B}\ten{A}_{(j)}$ we get
\begin{align*} \begin{align*}
\ten{X}_{(j)}\t{\ten{X}_{(j)}} \overset{\text{SVD}}{=} \mat{U}_j\mat{D}_j\t{\mat{U}_j} \ten{X}_{(j)}\t{\ten{X}_{(j)}} \overset{\text{SVD}}{=} \mat{U}_j\mat{D}_j\t{\mat{U}_j}
= \mat{\alpha}_j(\ten{F}\times_{i\in[r]\backslash j}\mat{\alpha}_i)_{(j)} = \mat{\alpha}_j(\ten{F}\times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}
\t{(\ten{F}\times_{i\in[r]\backslash j}\mat{\alpha}_i)_{(j)}}\t{\mat{\alpha}_j} \t{(\ten{F}\times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}}\t{\mat{\alpha}_j}
\end{align*} \end{align*}
for the $j = 1, ..., r$ modes. Using this relation we construct an iterative estimation process by setting the initial estimates of $\hat{\mat{\alpha}}_j^{(0)} = \mat{U}_j[, 1:q_j]$ which are the first $q_j$ columns of $\mat{U}_j$. for the $j = 1, ..., r$ modes. Using this relation we construct an iterative estimation process by setting the initial estimates of $\hat{\mat{\alpha}}_j^{(0)} = \mat{U}_j[, 1:q_j]$ which are the first $q_j$ columns of $\mat{U}_j$.
\todo{continue} For getting least squares estimates for $\mat{\alpha}_j$, $j = 1, ..., r$ we observe that by matricization of the mean model
\begin{displaymath}
\ten{X}_{(j)} = (\ten{F}\times_{k\in[r]}\mat{\alpha}_k)_{(j)} = \mat{\alpha}_j(\ten{F}\times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}
\end{displaymath}
leads to normal equations for each $\mat{\alpha}_j$, $j = 1, ..., r$
\begin{displaymath}
\ten{X}_{(j)}\t{(\ten{F}\times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}} = \mat{\alpha}_j(\ten{F}\times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}\t{(\ten{F}\times_{k\in[r]\backslash j}\mat{\alpha}_k)_{(j)}}
\end{displaymath}
where the normal equations for $\mat{\alpha}_j$ depend on all the other $\mat{\alpha}_k$. With the initial estimates from above this allows an alternating approach. Index with $t = 1, ...$ the current iteration, then a new estimate $\widehat{\mat{\alpha}}_j^{(t)}$ given the previous estimates $\widehat{\mat{\alpha}}_k^{(t-1)}$, $k = 1, ..., r$ is computed as
\begin{displaymath}
\widehat{\mat{\alpha}}_j^{(t)} =
\ten{X}_{(j)}
\t{\big(\ten{F}\times_{k\in[r]\backslash j}\widehat{\mat{\alpha}}_k^{(t-1)}\big)_{(j)}}
\left(
\big(\ten{F}\times_{k\in[r]\backslash j}\widehat{\mat{\alpha}}_k^{(t-1)}\big)_{(j)}
\t{\big(\ten{F}\times_{k\in[r]\backslash j}\widehat{\mat{\alpha}}_k^{(t-1)}\big)_{(j)}}
\right)^{-1}
\end{displaymath}
for $j = 1, ..., r$ until convergence or a maximum number of iterations is exceeded. The final estimates are the least squares estimates by this procedure.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Numerical Examples %%% %%% Numerical Examples %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerical Examples} \section{Numerical Examples}
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 % 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
\begin{displaymath} % \begin{displaymath}
\mat{X} = \mat{\beta}\mat{f}_y \t{\mat{\alpha}} + \mat{\epsilon}, \qquad \vec{\mat{\epsilon}} \sim \mathbb{N}_{p q}(\mat{0}, \mat{\Delta}) % \mat{X} = \mat{\beta}\mat{f}_y \t{\mat{\alpha}} + \mat{\epsilon}, \qquad \vec{\mat{\epsilon}} \sim \mathbb{N}_{p q}(\mat{0}, \mat{\Delta})
\end{displaymath} % \end{displaymath}
where $\mat{\Delta}_{i j} = 0.5^{|i - j|}$ for $i, j = 1, ..., p q$. % where $\mat{\Delta}_{i j} = 0.5^{|i - j|}$ for $i, j = 1, ..., p q$.
\begin{figure} \begin{table}[!ht]
\centering \centering
\includegraphics{loss_Ex01.png} % see: https://en.wikibooks.org/wiki/LaTeX/Tables
\end{figure} \begin{tabular}{ll|r@{ }l *{3}{r@{.}l}}
\begin{figure} method & init
\centering & \multicolumn{2}{c}{loss}
\includegraphics{estimates_Ex01.png} & \multicolumn{2}{c}{MSE}
\end{figure} & \multicolumn{2}{c}{$\dist(\hat{\mat\alpha}, \mat\alpha)$}
\begin{figure} & \multicolumn{2}{c}{$\dist(\hat{\mat\beta}, \mat\beta)$}
\centering \\ \hline
\includegraphics{Delta_Ex01.png} base & vlp & -2642&(1594) & 1&82 (2.714) & 0&248 (0.447) & 0&271 (0.458) \\
\end{figure} new & vlp & -2704&(1452) & 1&78 (2.658) & 0&233 (0.438) & 0&260 (0.448) \\
\begin{figure} new & ls & -3479& (95) & 0&99 (0.025) & 0&037 (0.017) & 0&035 (0.015) \\
\centering momentum & vlp & -2704&(1452) & 1&78 (2.658) & 0&233 (0.438) & 0&260 (0.448) \\
\includegraphics{hist_Ex01.png} momentum & ls & -3479& (95) & 0&99 (0.025) & 0&037 (0.017) & 0&035 (0.015) \\
\end{figure} approx & vlp & 6819&(1995) & 3&99 (12.256) & 0&267 (0.448) & 0&287 (0.457) \\
approx & ls & 5457& (163) & 0&99 (0.025) & 0&033 (0.017) & 0&030 (0.012) \\
\end{tabular}
\caption{Mean (standard deviation) for simulated runs of $20$ repititions for the model $\mat{X} = \mat{\beta}\mat{f}_y\t{\mat{\alpha}}$ of dimensinos $(p, q) = (11, 7)$, $(k, r) = (3, 5)$ 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.}
\end{table}
% \begin{figure}
% \centering
% \includegraphics{loss_Ex01.png}
% \end{figure}
% \begin{figure}
% \centering
% \includegraphics{estimates_Ex01.png}
% \end{figure}
% \begin{figure}
% \centering
% \includegraphics{Delta_Ex01.png}
% \end{figure}
% \begin{figure}
% \centering
% \includegraphics{hist_Ex01.png}
% \end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Bib and Index %%% %%% Bib and Index %%%
@ -707,41 +753,33 @@ Therefore, we use $\mat{A}, \mat{B}, \mat{X}, \mat{F}, \mat{R}, ...$ for matrice
\ten{T}\ttm[2]\mat{B}\ &{\widehat{=}}\ \mat{B}\ten{T}_{(2)} & \ten{T}(n, p, q), \ten{T}_{(2)}(p, n q), \mat{B}(q, p) \ten{T}\ttm[2]\mat{B}\ &{\widehat{=}}\ \mat{B}\ten{T}_{(2)} & \ten{T}(n, p, q), \ten{T}_{(2)}(p, n q), \mat{B}(q, p)
\end{align*} \end{align*}
\section{Matrix Valued Normal Distribution} % \section{Matrix Valued Normal Distribution}
A random variable $\mat{X}$ of dimensions $p\times q$ is \emph{Matrix-Valued Normal Distribution}, denoted % A random variable $\mat{X}$ of dimensions $p\times q$ is \emph{Matrix-Valued Normal Distribution}, denoted
\begin{displaymath} % \begin{displaymath}
\mat{X}\sim\mathcal{MN}_{p\times q}(\mat{\mu}, \mat{\Delta}_2, \mat{\Delta}_1), % \mat{X}\sim\mathcal{MN}_{p\times q}(\mat{\mu}, \mat{\Delta}_2, \mat{\Delta}_1),
\end{displaymath} % \end{displaymath}
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 % 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} % \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). % 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} % \end{displaymath}
\section{Sampling form a Multi-Array Normal Distribution} % \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 % 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
\begin{displaymath} % \begin{displaymath}
\ten{X}\sim\mathcal{TN}(\mu, \mat{\Delta}_1, ..., \mat{\Delta}_r). % \ten{X}\sim\mathcal{TN}(\mu, \mat{\Delta}_1, ..., \mat{\Delta}_r).
\end{displaymath} % \end{displaymath}
Its density is given by % Its density is given by
\begin{displaymath} % \begin{displaymath}
f(\ten{X}) = \Big( \prod_{i = 1}^r \sqrt{(2\pi)^{p_i}|\mat{\Delta}_i|^{q_i}} \Big)^{-1} % f(\ten{X}) = \Big( \prod_{i = 1}^r \sqrt{(2\pi)^{p_i}|\mat{\Delta}_i|^{q_i}} \Big)^{-1}
\exp\!\left( -\frac{1}{2}\langle \ten{X} - \mu, (\ten{X} - \mu)\times\{\mat{\Delta}_1^{-1}, ..., \mat{\Delta}_r^{-1}\} \rangle \right) % \exp\!\left( -\frac{1}{2}\langle \ten{X} - \mu, (\ten{X} - \mu)\times\{\mat{\Delta}_1^{-1}, ..., \mat{\Delta}_r^{-1}\} \rangle \right)
\end{displaymath} % \end{displaymath}
with $q_i = \prod_{j \neq i}p_j$. This is equivalent to the vectorized $\vec\ten{X}$ following a Multi-Variate Normal distribution % with $q_i = \prod_{j \neq i}p_j$. This is equivalent to the vectorized $\vec\ten{X}$ following a Multi-Variate Normal distribution
\begin{displaymath} % \begin{displaymath}
\vec{\ten{X}}\sim\mathcal{N}_{p}(\vec{\mu}, \mat{\Delta}_r\otimes...\otimes\mat{\Delta}_1) % \vec{\ten{X}}\sim\mathcal{N}_{p}(\vec{\mu}, \mat{\Delta}_r\otimes...\otimes\mat{\Delta}_1)
\end{displaymath} % \end{displaymath}
with $p = \prod_{i = 1}^r p_i$. % with $p = \prod_{i = 1}^r p_i$.
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 % \todo{Check this!!!}
\begin{displaymath}
\ten{Z}\sim\mathcal{TN}(0, \mat{I}_{p_1}, ..., \mat{I}_{p_r})
\quad\Rightarrow\quad
\ten{X} = \ten{Z}\times\{\mat{\Delta}_1^{1/2}, ..., \mat{\Delta}_r^{1/2}\} + \mu\sim\mathcal{TN}(\mu, \mat{\Delta}_1, ..., \mat{\Delta}_r).
\end{displaymath}
where the sampling from the standard Multi-Array Normal is done by sampling all of the elements of $\ten{Z}$ from a standard Normal.
\todo{Check this!!!}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Reference Summaries %%% %%% Reference Summaries %%%

View File

@ -18,17 +18,27 @@ sim <- function(X, Fy, alpha.true, beta.true, max.iter = 500L) {
# Logger creator # Logger creator
logger <- function(name) { logger <- function(name) {
eval(substitute(function(iter, loss, alpha, beta, ...) { eval(substitute(function(iter, loss, alpha, beta, ...) {
hist[iter + 1L, ] <<- c( tryCatch({
iter = iter, hist[iter + 1L, ] <<- c(
loss = loss, iter = iter,
dist = (dist <- dist.subspace(c(kronecker(alpha.true, beta.true)), loss = loss,
c(kronecker(alpha, beta)))), dist = (dist <- dist.subspace(c(kronecker(alpha.true, beta.true)),
dist.alpha = (dist.alpha <- dist.subspace(c(alpha.true), c(alpha))), c(kronecker(alpha, beta)))),
dist.beta = (dist.beta <- dist.subspace(c( beta.true), c(beta ))), dist.alpha = (dist.alpha <- dist.subspace(c(alpha.true), c(alpha))),
norm.alpha = norm(alpha, "F"), dist.beta = (dist.beta <- dist.subspace(c( beta.true), c(beta ))),
norm.beta = norm(beta, "F"), norm.alpha = norm(alpha, "F"),
mse = mean((X - mlm(Fy, alpha, beta, modes = 3:2))^2) norm.beta = norm(beta, "F"),
) mse = mean((X - mlm(Fy, alpha, beta, modes = 3:2))^2)
)},
error = function(e) {
cat("Error in ", name,
", dim(alpha): ", dim(alpha),
", dim(alpha.true): ", dim(alpha.true),
", dim(beta)", dim(beta),
", dim(beta.true)", dim(beta.true),
"\n")
stop(e)
})
cat(sprintf( cat(sprintf(
"%s(%3d) | l = %-12.4f - dist = %-.4e - alpha(%d, %d) = %-.4e - beta(%d, %d) = %-.4e\n", "%s(%3d) | l = %-12.4f - dist = %-.4e - alpha(%d, %d) = %-.4e - beta(%d, %d) = %-.4e\n",
@ -40,7 +50,11 @@ sim <- function(X, Fy, alpha.true, beta.true, max.iter = 500L) {
} }
# Initialize logger history targets # Initialize logger history targets
hist.base <- hist.new <- hist.momentum <- hist.approx <- hist.ls <- hist.base <-
hist.new.vlp <- hist.new.ls <-
hist.ls <-
hist.momentum.vlp <- hist.momentum.ls <-
hist.approx.vlp <- hist.approx.ls <-
data.frame(iter = seq(0L, max.iter), data.frame(iter = seq(0L, max.iter),
loss = NA, dist = NA, dist.alpha = NA, dist.beta = NA, loss = NA, dist = NA, dist.alpha = NA, dist.beta = NA,
norm.alpha = NA, norm.beta = NA, mse = NA norm.alpha = NA, norm.beta = NA, mse = NA
@ -49,53 +63,81 @@ sim <- function(X, Fy, alpha.true, beta.true, max.iter = 500L) {
# Base (old) # Base (old)
kpir.base(X, Fy, max.iter = max.iter, logger = logger("base")) kpir.base(X, Fy, max.iter = max.iter, logger = logger("base"))
# New (simple Gradient Descent) # New (simple Gradient Descent, using VLP initialization)
kpir.new(X, Fy, max.iter = max.iter, logger = logger("new")) kpir.new(X, Fy, max.iter = max.iter, init.method = "vlp",
logger = logger("new.vlp"))
kpir.new(X, Fy, max.iter = max.iter, init.method = "ls",
logger = logger("new.ls"))
# Least Squares estimate (alternating estimation) # Least Squares estimate (alternating estimation)
kpir.ls(X, Fy, sample.mode = 1L, max.iter = max.iter, logger = logger("ls")) kpir.ls(X, Fy, sample.mode = 1L, max.iter = max.iter, logger = logger("ls"))
# Gradient Descent with Nesterov Momentum # Gradient Descent with Nesterov Momentum
kpir.momentum(X, Fy, max.iter = max.iter, logger = logger("momentum")) kpir.momentum(X, Fy, max.iter = max.iter, init.method = "vlp",
logger = logger("momentum.vlp"))
kpir.momentum(X, Fy, max.iter = max.iter, init.method = "ls",
logger = logger("momentum.ls"))
# Approximated MLE with Nesterov Momentum # Approximated MLE with Nesterov Momentum
kpir.approx(X, Fy, max.iter = max.iter, logger = logger("approx")) kpir.approx(X, Fy, max.iter = max.iter, init.method = "vlp",
logger = logger("approx.vlp"))
kpir.approx(X, Fy, max.iter = max.iter, init.method = "ls",
logger = logger("approx.ls"))
# Add method tags # Add method tags
hist.base$method <- factor("base") hist.base$method <- factor("base")
hist.new$method <- factor("new") hist.new.vlp$method <- factor("new")
hist.ls$method <- factor("ls") hist.new.ls$method <- factor("new")
hist.momentum$method <- factor("momentum") hist.ls$method <- factor("ls")
hist.approx$method <- factor("approx") hist.momentum.vlp$method <- factor("momentum")
hist.momentum.ls$method <- factor("momentum")
hist.approx.vlp$method <- factor("approx")
hist.approx.ls$method <- factor("approx")
# Add init. method tag
hist.base$init <- factor("vlp")
hist.new.vlp$init <- factor("vlp")
hist.new.ls$init <- factor("ls")
hist.ls$init <- factor("ls")
hist.momentum.vlp$init <- factor("vlp")
hist.momentum.ls$init <- factor("ls")
hist.approx.vlp$init <- factor("vlp")
hist.approx.ls$init <- factor("ls")
# Combine results and return # Combine results and return
rbind(hist.base, hist.new, hist.momentum, hist.approx, hist.ls) rbind(
hist.base,
hist.new.vlp, hist.new.ls,
hist.ls,
hist.momentum.vlp, hist.momentum.ls,
hist.approx.vlp, hist.approx.ls
)
} }
## Plot helper function ## Plot helper function
plot.hist2 <- function(hist, response, type = "all", ...) { plot.hist2 <- function(hist, response, type = "all", ...) {
# Extract final results from history # Extract final results from history
sub <- na.omit(hist[c("iter", response, "method", "repetition")]) sub <- na.omit(hist[c("iter", response, "method", "init", "repetition")])
sub <- aggregate(sub, list(sub$method, sub$repetition), tail, 1) sub <- aggregate(sub, list(sub$method, sub$init, sub$repetition), tail, 1)
# Setup ggplot # Setup ggplot
p <- ggplot(hist, aes_(x = quote(iter), p <- ggplot(hist, aes_(x = quote(iter),
y = as.name(response), y = as.name(response),
color = quote(method), color = quote(method),
group = quote(interaction(method, repetition)))) linetype = quote(init),
group = quote(interaction(method, repetition, init))))
# Add requested layers # Add requested layers
if (type == "all") { if (type == "all") {
p <- p + geom_line(na.rm = TRUE) p <- p + geom_line(na.rm = TRUE)
p <- p + geom_point(data = sub) p <- p + geom_point(data = sub)
} else if (type == "mean") { } else if (type == "mean") {
p <- p + geom_line(alpha = 0.5, na.rm = TRUE, linetype = "dotted") p <- p + geom_line(alpha = 0.4, na.rm = TRUE, linetype = "dotted")
p <- p + geom_point(data = sub, alpha = 0.5) p <- p + geom_point(data = sub, alpha = 0.4)
p <- p + geom_line(aes(group = method), p <- p + geom_line(aes(group = interaction(method, init)),
stat = "summary", fun = "mean", na.rm = TRUE) stat = "summary", fun = "mean", na.rm = TRUE)
} else if (type == "median") { } else if (type == "median") {
p <- p + geom_line(alpha = 0.5, na.rm = TRUE, linetype = "dotted") p <- p + geom_line(alpha = 0.4, na.rm = TRUE, linetype = "dotted")
p <- p + geom_point(data = sub, alpha = 0.5) p <- p + geom_point(data = sub, alpha = 0.4)
p <- p + geom_line(aes(group = method), p <- p + geom_line(aes(group = interaction(method, init)),
stat = "summary", fun = "median", na.rm = TRUE) stat = "summary", fun = "median", na.rm = TRUE)
} }
# return with theme and annotations # return with theme and annotations
@ -182,7 +224,7 @@ max.iter <- 2
Delta.1 <- sqrt(0.5)^abs(outer(seq_len(p), seq_len(p), `-`)) Delta.1 <- sqrt(0.5)^abs(outer(seq_len(p), seq_len(p), `-`))
Delta.2 <- sqrt(0.5)^abs(outer(seq_len(q), seq_len(q), `-`)) Delta.2 <- sqrt(0.5)^abs(outer(seq_len(q), seq_len(q), `-`))
for (rep in 1:reps) { for (rep in 1:reps) {
cat(sprintf("%4d / %d simulation rep. started\n", rep, reps)) cat(sprintf("\n\033[1m%4d / %d simulation rep. started\033[0m\n", rep, reps))
alpha.1.true <- alpha.1 <- matrix(rnorm(q * r), q, r) alpha.1.true <- alpha.1 <- matrix(rnorm(q * r), q, r)
alpha.2.true <- alpha.2 <- matrix(rnorm(p * k), p, k) alpha.2.true <- alpha.2 <- matrix(rnorm(p * k), p, k)
@ -194,10 +236,10 @@ for (rep in 1:reps) {
head(rep(c(0, pi / 2), ceiling(0.5 * k * r)), k * r) head(rep(c(0, pi / 2), ceiling(0.5 * k * r)), k * r)
)) ))
dim(Fy) <- c(n, k, r) dim(Fy) <- c(n, k, r)
X <- mlm(Fy, alpha, beta, modes = 3:2) X <- mlm(Fy, alpha.1, alpha.2, modes = 3:2)
X <- X + rtensornorm(n, 0, Delta.1, Delta.2, sample.mode = 1L) X <- X + rtensornorm(n, 0, Delta.1, Delta.2, sample.mode = 1L)
hist.sim <- sim(X, Fy, alpha.true, beta.true, max.iter = max.iter) hist.sim <- sim(X, Fy, alpha.1.true, alpha.2.true, max.iter = max.iter)
hist.sim$repetition <- rep hist.sim$repetition <- rep
hist <- rbind(hist, hist.sim) hist <- rbind(hist, hist.sim)
@ -213,17 +255,47 @@ hist$repetition <- factor(hist$repetition)
for (response in c("loss", "mse", "dist", "dist.alpha", "dist.beta")) { for (response in c("loss", "mse", "dist", "dist.alpha", "dist.beta")) {
for (fun in c("all", "mean", "median")) { for (fun in c("all", "mean", "median")) {
print(plot.hist2(hist, response, fun, title = fun) + coord_trans(x = "log1p")) title <- paste(fun, paste(c("n", "p", "q", "k", "r"), c(n, p, q, k, r), sep = "=", collapse = ", "))
print(plot.hist2(hist, response, fun, title = title) +
coord_trans(x = "log1p"))
dev.print(png, file = sprintf("%s_%s_%s_%s.png", sim.name, datetime, response, fun), dev.print(png, file = sprintf("%s_%s_%s_%s.png", sim.name, datetime, response, fun),
width = 768, height = 768, res = 125) width = 768, height = 768, res = 125)
if (response != "loss") { if (response != "loss") {
print(plot.hist2(hist, response, fun, title = fun) + coord_trans(x = "log1p", y = "log1p")) print(plot.hist2(hist, response, fun, title = title) +
coord_trans(x = "log1p", y = "log1p"))
dev.print(png, file = sprintf("%s_%s_%s_%s_log.png", sim.name, datetime, response, fun), dev.print(png, file = sprintf("%s_%s_%s_%s_log.png", sim.name, datetime, response, fun),
width = 768, height = 768, res = 125) width = 768, height = 768, res = 125)
} }
} }
} }
stats <- local({
# final result from history
sub <- na.omit(hist)
sub <- aggregate(sub, list(
method = sub$method, init = sub$init, repetition = sub$repetition
), tail, 1)
# aggregate statistics over repetitions
stats.mean <- aggregate(subset(sub, select = c("loss", "mse", "dist.alpha", "dist.beta")),
list(method = sub$method, init = sub$init), mean)
stats.sd <- aggregate(subset(sub, select = c("loss", "mse", "dist.alpha", "dist.beta")),
list(method = sub$method, init = sub$init), sd)
# merge mean and sd stats together
merge(stats.mean, stats.sd, by = c("method", "init"), suffixes = c(".mean", ".sd"))
})
print(stats, digits = 2)
# method init loss.mean mse.mean dist.alpha.mean dist.beta.mean loss.sd mse.sd dist.alpha.sd dist.beta.sd
# 1 approx ls 5457 0.99 0.033 0.030 163 0.025 0.017 0.012
# 2 approx vlp 6819 3.99 0.267 0.287 1995 12.256 0.448 0.457
# 3 base vlp -2642 1.82 0.248 0.271 1594 2.714 0.447 0.458
# 4 momentum ls -3479 0.99 0.037 0.035 95 0.025 0.017 0.015
# 5 momentum vlp -2704 1.78 0.233 0.260 1452 2.658 0.438 0.448
# 6 new ls -3479 0.99 0.037 0.035 95 0.025 0.017 0.015
# 7 new vlp -2704 1.78 0.233 0.260 1452 2.658 0.438 0.448
################################################################################ ################################################################################
### Sim 3 ### ### Sim 3 ###
################################################################################ ################################################################################

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@ -15,7 +15,6 @@ export(dist.projection)
export(dist.subspace) export(dist.subspace)
export(kpir.approx) export(kpir.approx)
export(kpir.base) export(kpir.base)
export(kpir.kron)
export(kpir.ls) export(kpir.ls)
export(kpir.momentum) export(kpir.momentum)
export(kpir.new) export(kpir.new)

View File

@ -25,7 +25,7 @@
#' dataset <- function(name, n = 60, p = 24) { #' dataset <- function(name, n = 60, p = 24) {
#' name <- toupper(name) #' name <- toupper(name)
#' if (!startsWith('M', name)) { name <- paste0('M', name) } #' if (!startsWith('M', name)) { name <- paste0('M', name) }
#' #'
#' if (name %in% c('M1', 'M2', 'M3')) { #' if (name %in% c('M1', 'M2', 'M3')) {
#' Sigma <- 0.5^abs(outer(1:p, 1:p, `-`)) #' Sigma <- 0.5^abs(outer(1:p, 1:p, `-`))
#' X <- rmvnorm(n, sigma = Sigma) #' X <- rmvnorm(n, sigma = Sigma)
@ -52,7 +52,7 @@
#' } else { #' } else {
#' stop('Unknown dataset name.') #' stop('Unknown dataset name.')
#' } #' }
#' #'
#' list(X = X, y = y, B = B) #' list(X = X, y = y, B = B)
#' } #' }
#' # Sample dataset #' # Sample dataset
@ -63,16 +63,16 @@
#' P.Fy <- Fy %*% solve(crossprod(Fy), t(Fy)) #' P.Fy <- Fy %*% solve(crossprod(Fy), t(Fy))
#' M <- with(ds, crossprod(X, P.Fy %*% X) / nrow(X)) # Sigma Fit #' M <- with(ds, crossprod(X, P.Fy %*% X) / nrow(X)) # Sigma Fit
#' N <- cov(ds$X) # Sigma #' N <- cov(ds$X) # Sigma
#' #'
#' fits <- CISE(M, N, d = ncol(ds$B), Theta = log(seq(1, exp(1e-3), len = 1000))) #' fits <- CISE(M, N, d = ncol(ds$B), Theta = log(seq(1, exp(1e-3), len = 1000)))
#' #'
#' BIC <- unlist(Map(attr, fits, 'BIC')) #' BIC <- unlist(Map(attr, fits, 'BIC'))
#' df <- unlist(Map(attr, fits, 'df')) #' df <- unlist(Map(attr, fits, 'df'))
#' dist <- unlist(Map(attr, fits, 'dist')) #' dist <- unlist(Map(attr, fits, 'dist'))
#' iter <- unlist(Map(attr, fits, 'iter')) #' iter <- unlist(Map(attr, fits, 'iter'))
#' theta <- unlist(Map(attr, fits, 'theta')) #' theta <- unlist(Map(attr, fits, 'theta'))
#' p.theta <- unlist(Map(function(V) sum(rowSums(V^2) > 1e-9), fits)) #' p.theta <- unlist(Map(function(V) sum(rowSums(V^2) > 1e-9), fits))
#' #'
#' par(mfrow = c(2, 2)) #' par(mfrow = c(2, 2))
#' plot(theta, BIC, type = 'l') #' plot(theta, BIC, type = 'l')
#' plot(theta, p.theta, type = 'l') #' plot(theta, p.theta, type = 'l')
@ -84,7 +84,9 @@
#' Reduction and Variable Selection" By Xin Chen, Changliang Zou and #' Reduction and Variable Selection" By Xin Chen, Changliang Zou and
#' R. Dennis Cook. #' R. Dennis Cook.
#' #'
#' @suggest RSpectra #' @note for speed reasons this functions attempts to use
#' \code{\link[RSpectra]{eigs_sym}} if \pkg{\link{RSpectra}} is installed,
#' otherwise \code{\link{eigen}} is used which might be significantly slower.
#' #'
#' @export #' @export
CISE <- function(M, N, d = 1L, method = "PFC", max.iter = 100L, Theta = NULL, CISE <- function(M, N, d = 1L, method = "PFC", max.iter = 100L, Theta = NULL,

View File

@ -9,6 +9,7 @@
kpir.approx <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])), kpir.approx <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
max.iter = 500L, max.line.iter = 50L, step.size = 1e-3, max.iter = 500L, max.line.iter = 50L, step.size = 1e-3,
nesterov.scaling = function(a, t) 0.5 * (1 + sqrt(1 + (2 * a)^2)), nesterov.scaling = function(a, t) 0.5 * (1 + sqrt(1 + (2 * a)^2)),
max.init.iter = 20L, init.method = c("ls", "vlp"),
eps = .Machine$double.eps, eps = .Machine$double.eps,
logger = NULL logger = NULL
) { ) {
@ -17,7 +18,7 @@ kpir.approx <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
stopifnot(nrow(X) == NROW(Fy)) stopifnot(nrow(X) == NROW(Fy))
n <- nrow(X) # Number of observations n <- nrow(X) # Number of observations
# Get and check predictor dimensions # Check predictor dimensions
if (length(dim(X)) == 2L) { if (length(dim(X)) == 2L) {
stopifnot(!missing(shape)) stopifnot(!missing(shape))
stopifnot(ncol(X) == prod(shape[1:2])) stopifnot(ncol(X) == prod(shape[1:2]))
@ -26,18 +27,16 @@ kpir.approx <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
} else if (length(dim(X)) == 3L) { } else if (length(dim(X)) == 3L) {
p <- dim(X)[2] p <- dim(X)[2]
q <- dim(X)[3] q <- dim(X)[3]
dim(X) <- c(n, p * q)
} else { } else {
stop("'X' must be a matrix or 3-tensor") stop("'X' must be a matrix or 3-tensor")
} }
# Get and check response dimensions # Check response dimensions
if (!is.array(Fy)) { if (!is.array(Fy)) {
Fy <- as.array(Fy) Fy <- as.array(Fy)
} }
if (length(dim(Fy)) == 1L) { if (length(dim(Fy)) == 1L) {
k <- r <- 1L k <- r <- 1L
dim(Fy) <- c(n, 1L)
} else if (length(dim(Fy)) == 2L) { } else if (length(dim(Fy)) == 2L) {
stopifnot(!missing(shape)) stopifnot(!missing(shape))
stopifnot(ncol(Fy) == prod(shape[3:4])) stopifnot(ncol(Fy) == prod(shape[3:4]))
@ -46,34 +45,44 @@ kpir.approx <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
} else if (length(dim(Fy)) == 3L) { } else if (length(dim(Fy)) == 3L) {
k <- dim(Fy)[2] k <- dim(Fy)[2]
r <- dim(Fy)[3] r <- dim(Fy)[3]
dim(Fy) <- c(n, k * r)
} else { } else {
stop("'Fy' must be a vector, matrix or 3-tensor") stop("'Fy' must be a vector, matrix or 3-tensor")
} }
### Step 1: (Approx) Least Squares solution for `X = Fy B' + epsilon` ### Step 1: (Approx) Least Squares initial estimate
# Vectorize init.method <- match.arg(init.method)
dim(Fy) <- c(n, k * r) if (init.method == "ls") {
dim(X) <- c(n, p * q) # De-Vectroize (from now on tensor arithmetics)
# Solve dim(Fy) <- c(n, k, r)
cpFy <- crossprod(Fy) # TODO: Check/Test and/or replace dim(X) <- c(n, p, q)
if (n <= k * r || qr(cpFy)$rank < k * r) {
# In case of under-determined system replace the inverse in the normal ls <- kpir.ls(X, Fy, max.iter = max.init.iter, sample.mode = 1L, eps = eps)
# equation by the Moore-Penrose Pseudo Inverse c(beta0, alpha0) %<-% ls$alphas
B <- t(matpow(cpFy, -1) %*% crossprod(Fy, X)) } else { # Van Loan and Pitsianis
} else { # Vectorize
# Compute OLS estimate by the Normal Equation dim(Fy) <- c(n, k * r)
B <- t(solve(cpFy, crossprod(Fy, X))) dim(X) <- c(n, p * q)
# solution for `X = Fy B' + epsilon`
cpFy <- crossprod(Fy) # TODO: Check/Test and/or replace
if (n <= k * r || qr(cpFy)$rank < k * r) {
# In case of under-determined system replace the inverse in the normal
# equation by the Moore-Penrose Pseudo Inverse
B <- t(matpow(cpFy, -1) %*% crossprod(Fy, X))
} else {
# Compute OLS estimate by the Normal Equation
B <- t(solve(cpFy, crossprod(Fy, X)))
}
# De-Vectroize (from now on tensor arithmetics)
dim(Fy) <- c(n, k, r)
dim(X) <- c(n, p, q)
# Decompose `B = alpha x beta` into `alpha` and `beta`
c(alpha0, beta0) %<-% approx.kronecker(B, c(q, r), c(p, k))
} }
# De-Vectroize (from now on tensor arithmetics)
dim(Fy) <- c(n, k, r)
dim(X) <- c(n, p, q)
# Decompose `B = alpha x beta` into `alpha` and `beta`
c(alpha0, beta0) %<-% approx.kronecker(B, c(q, r), c(p, k))
# Compute residuals # Compute residuals
R <- X - (Fy %x_3% alpha0 %x_2% beta0) R <- X - (Fy %x_3% alpha0 %x_2% beta0)

View File

@ -5,6 +5,7 @@
kpir.momentum <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])), kpir.momentum <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
max.iter = 500L, max.line.iter = 50L, step.size = 1e-3, max.iter = 500L, max.line.iter = 50L, step.size = 1e-3,
nesterov.scaling = function(a, t) 0.5 * (1 + sqrt(1 + (2 * a)^2)), nesterov.scaling = function(a, t) 0.5 * (1 + sqrt(1 + (2 * a)^2)),
max.init.iter = 20L, init.method = c("ls", "vlp"),
eps = .Machine$double.eps, eps = .Machine$double.eps,
logger = NULL logger = NULL
) { ) {
@ -48,19 +49,30 @@ kpir.momentum <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
} }
### Step 1: (Approx) Least Squares solution for `X = Fy B' + epsilon` ### Step 1: (Approx) Least Squares initial estimate
cpFy <- crossprod(Fy) # TODO: Check/Test and/or replace init.method <- match.arg(init.method)
if (n <= k * r || qr(cpFy)$rank < k * r) { if (init.method == "ls") {
# In case of under-determined system replace the inverse in the normal dim(X) <- c(n, p, q)
# equation by the Moore-Penrose Pseudo Inverse dim(Fy) <- c(n, k, r)
B <- t(matpow(cpFy, -1) %*% crossprod(Fy, X)) ls <- kpir.ls(X, Fy, max.iter = max.init.iter, sample.mode = 1L, eps = eps)
} else { c(beta0, alpha0) %<-% ls$alphas
# Compute OLS estimate by the Normal Equation dim(X) <- c(n, p * q)
B <- t(solve(cpFy, crossprod(Fy, X))) dim(Fy) <- c(n, k * r)
} } else { # Van Loan and Pitsianis
# solution for `X = Fy B' + epsilon`
cpFy <- crossprod(Fy) # TODO: Check/Test and/or replace
if (n <= k * r || qr(cpFy)$rank < k * r) {
# In case of under-determined system replace the inverse in the normal
# equation by the Moore-Penrose Pseudo Inverse
B <- t(matpow(cpFy, -1) %*% crossprod(Fy, X))
} else {
# Compute OLS estimate by the Normal Equation
B <- t(solve(cpFy, crossprod(Fy, X)))
}
# Decompose `B = alpha x beta` into `alpha` and `beta` # Decompose `B = alpha x beta` into `alpha` and `beta`
c(alpha0, beta0) %<-% approx.kronecker(B, c(q, r), c(p, k)) c(alpha0, beta0) %<-% approx.kronecker(B, c(q, r), c(p, k))
}
# Compute residuals # Compute residuals
resid <- X - tcrossprod(Fy, kronecker(alpha0, beta0)) resid <- X - tcrossprod(Fy, kronecker(alpha0, beta0))

View File

@ -3,6 +3,7 @@
#' @export #' @export
kpir.new <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])), kpir.new <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
max.iter = 500L, max.line.iter = 50L, step.size = 1e-3, max.iter = 500L, max.line.iter = 50L, step.size = 1e-3,
max.init.iter = 20L, init.method = c("ls", "vlp"),
eps = .Machine$double.eps, eps = .Machine$double.eps,
logger = NULL logger = NULL
) { ) {
@ -45,19 +46,30 @@ kpir.new <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
stop("'Fy' must be a vector, matrix or 3-tensor") stop("'Fy' must be a vector, matrix or 3-tensor")
} }
### Step 1: (Approx) Least Squares solution for `X = Fy B' + epsilon` ### Step 1: (Approx) Least Squares initial estimate
cpFy <- crossprod(Fy) # TODO: Check/Test and/or replace init.method <- match.arg(init.method)
if (n <= k * r || qr(cpFy)$rank < k * r) { if (init.method == "ls") {
# In case of under-determined system replace the inverse in the normal dim(X) <- c(n, p, q)
# equation by the Moore-Penrose Pseudo Inverse dim(Fy) <- c(n, k, r)
B <- t(matpow(cpFy, -1) %*% crossprod(Fy, X)) ls <- kpir.ls(X, Fy, max.iter = max.init.iter, sample.mode = 1L, eps = eps)
} else { c(beta, alpha) %<-% ls$alphas
# Compute OLS estimate by the Normal Equation dim(X) <- c(n, p * q)
B <- t(solve(cpFy, crossprod(Fy, X))) dim(Fy) <- c(n, k * r)
} } else { # Van Loan and Pitsianis
# solution for `X = Fy B' + epsilon`
cpFy <- crossprod(Fy) # TODO: Check/Test and/or replace
if (n <= k * r || qr(cpFy)$rank < k * r) {
# In case of under-determined system replace the inverse in the normal
# equation by the Moore-Penrose Pseudo Inverse
B <- t(matpow(cpFy, -1) %*% crossprod(Fy, X))
} else {
# Compute OLS estimate by the Normal Equation
B <- t(solve(cpFy, crossprod(Fy, X)))
}
# Decompose `B = alpha x beta` into `alpha` and `beta` # Decompose `B = alpha x beta` into `alpha` and `beta`
c(alpha, beta) %<-% approx.kronecker(B, c(q, r), c(p, k)) c(alpha, beta) %<-% approx.kronecker(B, c(q, r), c(p, k))
}
# Compute residuals # Compute residuals
resid <- X - tcrossprod(Fy, kronecker(alpha, beta)) resid <- X - tcrossprod(Fy, kronecker(alpha, beta))