Vectorize and minimize Python loops$$

You$$must$$only use SVD $($i$.$e$.,$only use$$np$.$linalg.svd$).$

import numpy as np

import matplotlib.pyplot as plt

from sklearn.datasets import fetch$\_$olivetti$\_$faces

from sklearn.neighbors import KNeighborsClassifier

x$\_$train, y$\_$train, x$\_$test, y$\_$test $=$get$\_$faces$\_$data$($n$\_$train $=2\_000,$n$\_$test $=10\_000)$

$1.$Use SVD to find a basis of the matrix of the training data. The training data should be a $2$D array. Each row of the array should be an image flattened into a vector.

$2.$Visualize $5$or $6$basis vectors. $$Looking at the visualization $--$what does the$$most important$$basis vector represent? What do the next $3$vectors represent? Using def plot$\_$images$\_$in$\_$a$\_$row$($ims$)$:

$$

$\_,$axs $=$plt$.$subplots$(1,$len$($ims$),$figsize $=(15,5))$

$$for im$,$ax in zip$($ims$,$axs$.$ravel$())$:

$$ax$.$imshow$($im$,$cmap$=$'gray'$)$

$$ax$.$axis$(\text{'}$off$\text{'})$

$3.$Find a way to express any given image $($in particular from the test dataset$)$in your new basis. Find the simplest way possible. Now each image should be a vector of size$64^2.$Express$$all$$training and test images in this new basis

$4.$Pick $5$vectors$,$ each representing an image from the$$test dataset$$in the new basis. Write code which goes back to the standard basis and visualizes the images. Verify that the images you obtained this way are similar $($if not identical$)$to the originals. Plot them side by side.

$5.$Implement a function that expresses each image in a collection of images $($from the training dataset, test dataset, or any other images of the same shape$)$as a vector of$$r$$coordinates$.$The resulting vector must represent the coordinates in the$$r$$most prominent new basis vectors computed earlier. $($In other words these should be coordinates in a truncated basis$).$r$$should be a parameter of the function.The coordinates$$must$$come from the basis computed for the train dataset using SVD$.$

$6.$Pick$$r$-$dimensional vectors of $5$selected images from the$$test dataset. Visualize the images they represent $--$display each image for different values of$$r$.$Tune your choice of$$r$$based on what you see. Briefly describe your process of choosing the right$$r$.$

$7.$Perform your dimensionality reduction of the train and test dataset. Based on the observation from the previous part, choose your$$r$--$the number of basis vectors you will use. $$just use the information coming from SVD on the train data only

$8.$Construct$,$ train, test and tune your$$k$-$nearest neighbours classifier using the$$r$-$dimensional representation.train on the dimensionality$-$reduced training data and test on the dimensionality$-$reduced test data.use the data obtained from SVD$.$use the$$KNeighborsClassifier$$

$9.$Get$>90\%$accuracy on the$$test set$$and$$keep the running time on the entire test set below$1$second$.$Fix your final$$r$$and other parameters and train the classifier using the reduced data produced using these parameters.don't have to re$-$compute the SVD$.$Just recompute whatever depends on the parameter$$r