forked from SPC-education/numerics-2022
113 lines
4.7 KiB
TeX
113 lines
4.7 KiB
TeX
\documentclass[prb, notitlepage, aps, 10pt]{revtex4-2}
|
|
\usepackage[utf8]{inputenc}
|
|
\usepackage{listings}
|
|
\usepackage{amssymb}
|
|
\usepackage{graphicx,amsmath}
|
|
\usepackage{enumitem}
|
|
\usepackage{nicefrac}
|
|
\usepackage{amsmath}
|
|
\usepackage{graphicx}
|
|
\usepackage{amsfonts}
|
|
\usepackage{comment}
|
|
\usepackage{bm}
|
|
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
|
|
\usepackage{hyperref}
|
|
\hypersetup{
|
|
colorlinks=true,
|
|
linkcolor=blue,
|
|
filecolor=magenta,
|
|
urlcolor=cyan,
|
|
pdfpagemode=FullScreen,
|
|
}
|
|
|
|
\begin{document}
|
|
|
|
\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}
|
|
|
|
\bibliography{library.bib}
|
|
\end{document}
|