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kyunghyuncho committed Apr 19, 2017
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%\usepackage{bibentry}
%\nobibliography*

\begin{document}

\title{Homework Assignment 6}
\author{Lecturer: Kyunghyun Cho}
%\affil{
% Courant Institute of Mathematical Sciences and \\
% Center for Data Science,\\
% New York University
%}

\maketitle
\pagenumbering{arabic}

\paragraph{1.}

(a) Why is it more efficient to process data points if they are lower-dimensional
vectors? State one reason.

\vspace{5mm}

(b) What is a potential trouble of reducing the dimensionality of input vectors
before training a classifier? State one reason.

\vspace{5mm}

\paragraph{2.}

(a) Given a training set $D=\left\{ \vx_1, \ldots, \vx_N \right\}$, show that the
reconstruction error of principal component analysis (PCA) could be written down
as
\begin{align*}
\frac{1}{N} \sum_{n=1}^N \| \vx_n - \hat{\vx_n} \|^2_2 =
\sum_{j=q+1}^d \vw_i^\top \mC \vw_i,
\end{align*}
where $\vw_i$ is the $i$-th principal component or the eigenvector of the input
covariance matrix $\mC$.


\vspace{15mm}

(b) Show that
\begin{align*}
&\Sigma = \mW^\top \mC \mW \\
\iff&
\sigma_j^2 = \vw_j^\top \mC \vw_j,\text{ for all }j=1,\ldots,d,
\end{align*}
where $\mW$ is the weight matrix of PCA, $\mC$ is the input covariance
matrix, and
\begin{align*}
\Sigma = \diag(\sigma_1^2, \ldots, \sigma_q^2) =
\left[
\begin{array}{c c c c}
\sigma_1^2 & 0 & \cdots & 0 \\
0 & \sigma2^2 & \cdots & 0 \\
\vdots & 0 & \cdots & \vdots \\
\vdots & \vdots & \cdots & \vdots \\
0 & 0 & \cdots & \sigma_q^2
\end{array}
\right]
\end{align*}
is the covariance matrix of the code vectors.

\vspace{30mm}








%\bibliographystyle{abbrv}
%\bibliography{../lecture_note}


\end{document}






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