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\title{\texorpdfstring{Numerical Methods, Fall 2022\\ Assignment 2 [SVD decomposition with applications] \\ Total: 40, Deadline: 21 Oct}{}}

\maketitle

\section*{Suggested Reading}

\begin{itemize}
\item Lectures 4-5 of \cite{trefethen1997numerical}
\item Lecture 2 of \cite{tyrtyshnikov2012brief}
\item \href{https://stats.stackexchange.com/questions/2691/making-sense-of-principal-component-analysis-eigenvectors-eigenvalues}{Making sense of principal component analysis, eigenvectors and eigenvalues}
\end{itemize}

\section*{Exercises}

\begin{enumerate}

\begin{comment}
\item (5) Construct (manually) SVD decomposition of the following matrices:
$$
(a)\quad\begin{bmatrix}
3 & 0\\
0 & -2
\end{bmatrix},\quad
(b)\quad\begin{bmatrix}
0 & 2\\
0 & 0\\
0 & 0
\end{bmatrix},\quad
(c)\quad\begin{bmatrix}
1 & 1\\
1 & 1
\end{bmatrix}.
$$
For the case (b), construct both full and reduced SVD decomposition via \texttt{np.linalg.svd}. 
\end{comment}

\item (10) In this exercise, we will explore three main algorithms available in scientific \lstinline{python} distributions for computation of SVD: \lstinline{numpy.linalg.svd}, \lstinline{scipy.sparse.linalg.svds} and \lstinline{sklearn.utils.extmath.randomized_svd}. To this end:
\begin{itemize}
    \item Construct a random $n\times n$ matrix $A$ (with iid elements sampled from standard normal distribution); consider $n=2000$.
    \item Using these implementations, construct rank-2 approximations to $A$. You will thus obtain three rank-2 matrices $A_\textrm{svd}$, $A_\textrm{svds}$ and $A_\textrm{rsvd}$. Measure the run--time of these three algorithms for the given task.
    \item Compute the error norms: $\Vert A-A_\textrm{svd}\Vert_F$, $\Vert A-A_\textrm{svds}\Vert_F$, $\Vert A-A_\textrm{rsvd}\Vert_F$. Explain the result.
\end{itemize} 
\item (5) Let $A$ be $m\times n$ with SVD $A = U\Sigma V^T$. Compute SVDs of the following matrices in terms of $U$, $\Sigma$ and $V$: (i) $\left(A^T A\right)^{-1}$, (ii)  $\left(A^T A\right)^{-1}A^T$, (iii) $A\left(A^T A\right)^{-1}$, (iv) $A\left(A^T A\right)^{-1}A^T$.

\item (10) Consider the matrix:
$$
\begin{bmatrix}
-2 & 11\\
-10 & 5
\end{bmatrix}
$$
\begin{itemize}
    \item List the singular values, left singular vectors and right singular vectors of $A$. The SVD is not unique, so find the one that has the minimal number of minus signs in $U$ and $V$. 
    \item Draw a labeled picture of the unit ball in $\mathbb{R}^2$ and its image under $A$, together with the singular vectors with the coordinates of their vertices marked.
    \item What are $2$-norm and Frobenius norm of $A$?
    \item Find $A^{-1}$ not directly, but via SVD.
    \item Find the eigenvalues $\lambda_1, \lambda_2$ of $A$.
\end{itemize}
\item (5) The file \path{A.npy} contains the $n\times n$ matrix $A$. Determine the best approximation of $A_{ij}$ in terms of the following anzats, where the variables are separated: $A_{ij}\approx h_i\eta_j$. What is the related relative error of such approximation:
$$
\delta_{\textrm{err}}=\frac{\sqrt{\sum_{ij}\left(A_{ij}-h_i\eta_j\right)^2}}{\sqrt{\sum_{ij}A_{ij}^2}}?
$$
How many terms $K$ would an exact representation of the following form:
$$
A_{ij}=\sum_{\alpha=1}^K h_{\alpha i}\eta_{\alpha j}
$$
require?
\item (10) In this exercise, you will explore application of SVD to dimensionality reduction. Let us start with loading the dataset:
\lstset{language=Python}
\lstset{frame=lines}
\lstset{label={lst:code_direct}}
\lstset{basicstyle=\ttfamily}
\begin{lstlisting}
from sklearn.datasets import load_digits
digits = load_digits()
A = digits.data
y = digits.target
\end{lstlisting}
\end{enumerate}
so that rows of A contain monochromatic images of digits (64 float values which should be reshaped into $8\times 8$ images) and $y$ contains the digit labels.
\begin{itemize}
\item Inspect the dataset: plot examples of images, corresponding to several digits (say $0, 3, 7$).
\item Normalize the dataset $A$.
\item Use SVD to project the dataset $A$ from $64$ dimensions to $2$ dimensions. Show the colored scatter plot, where colors encode the digits.
\end{itemize}

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