Using the functions provided in the Jupyter notebook $$hw$-$pca$-$functions.ipynb$,$ run the principal component

analysis $($PCA$)$ on face data stored in the file called $$faces$.$mat$.$ Then, reduce the dimension of the sample

from $1024(32$ by $32)$ to $75$ and visualise the faces after the dimension reduction. Then, compute the

maximum and the L$2$ norm error between the original data set and the reduced data.

$$ Report all your results in a well$-$organized and described way. Submit the codes and the results.

hw$-$pca$-$functions.ipynb:

Use the functions below to solve the exercise

import numpy as np import scipy.io as sio import matplotlib.pyplot as plt

The feature$\_$normalize$()$ normalizes the features in the given dataset.

def feature$\_$normalize$($X$)$: $"""$ Normalizes the features in X$.$ Parameters $----------$ X : ndarray, shape $($n$\_$samples, n$\_$features$)$ Samples, where n$\_$samples is the number of samples and n$\_$features is the number of features. Returns $-------$ X$\_$norm : ndarray, shape $($n$\_$samples, n$\_$features$)$ Normalized training vectors. mu : ndarray, shape $($n$\_$feature, $)$ Mean value of each feature. sigma : ndarray, shape $($n$\_$feature, $)$ Standard deviation of each feature. $"""$ mu $=$ np$.$mean$($X$,$ axis$=0)$ sigma $=$ np$.$std$($X$,$ axis$=0,$ ddof$=1)$ X$\_$norm $=($X $-$ mu$)/$ sigma return X$\_$norm, mu$,$ sigma

The pca$()$ runs principal component analysis on the given dataset.

def pca$($X$)$: $"""$ Run principal component analysis on the dataset X$.$ Parameters $----------$ X : ndarray, shape $($n$\_$samples, n$\_$features$)$ Samples, where n$\_$samples is the number of samples and n$\_$features is the number of features. Returns $-------$ U : ndarray, shape $($n$\_$features, n$\_$features$)$ Unitary matrices. S : ndarray, shape $($n$\_$features,$)$ The singular values for every matrix. V : ndarray, shape $($n$\_$features, n$\_$features$)$ Unitary matrices. $"""$ m$,$ n $=$ X$.$shape sigma $=$ X$.$T$.$dot$($X$)/$ m U$,$ S$,$ V $=$ np$.$linalg.svd$($sigma$)$ return U$,$ S$,$ V

The project$\_$data$()$ projects the given data to the top K eigenvectors.

def project$\_$data$($X$,$ U$,$ K$)$: $"""$ Computes the reduced data representation when projecting only on to the top K eigenvectors. Parameters $----------$ X : ndarray, shape $($n$\_$samples, n$\_$features$)$ Samples, where n$\_$samples is the number of samples and n$\_$features is the number of features. U : ndarray, shape $($n$\_$features, n$\_$features$)$ Unitary matrices. K : int Reduced dimension. Returns $-------$ Z : ndarray, shape $($n$\_$samples, K$)$ The projection of X into the reduced dimensional space spanned by the first K columns of U$."""$ Z $=$ X$.$dot$($U$[$:$,0$:K$])$ return Z

The recover$\_$data$()$ recovers an approximation of the original data from the projected data.

def recover$\_$data$($Z$,$ U$,$ K$)$: $"""$ Recovers an approximation of the original data when using the projected data. Parameters $----------$ Z : ndarray, shape $($n$\_$samples, K$)$ The projected data, where n$\_$samples is the number of samples and K is the number of reduced dimensions. U : ndarray, shape $($n$\_$features, n$\_$features$)$ Unitary matrices, where n$\_$features is the number of features. K : int Reduced dimension. Returns $-------$ X$\_$rec : ndarray, shape $($n$\_$samples, n$\_$features$)$ The recovered samples. $"""$ X$\_$rec $=$ Z$.$dot$($U$[$:$,0$:K$].$T$)$ return X$\_$rec