Using this Matlab code:

_{clear all; close all; % clear workspace}

_{websave('mnist.mat',"https://github.com/daniel-e/mnist_octave/raw/master/mnist.mat");}

_{data=load('mnist.mat');}

_{X = double(data.trainX);}

_{[N,D]=size(X);}

_{% visualising the first image in MNIST dataset}

_{figure;imagesc(reshape(X(1,:), 28, 28)'),colormap("gray")}

_{% computing the mean image of MNIST dataset}

_{MeanImage=mean(X,1);}

_{figure;imagesc( reshape(MeanImage, 28, 28)'),colormap("gray")}

_{% centering data}

_{Xtilde=X-mean(X);}

_{% covariance}

_{S=Xtilde'*Xtilde/N;}

_{[U,L,V]=svd(S);}

_{% reconstruction of a test image}

_{xtest = double(data.testX(1,:));}

_{figure;imagesc(reshape(xtest, 28, 28)'),colormap("gray")}

_{% reconstruction using #nbVector eigenvectors}

_nbVector=10;

_{Reconstruction = MeanImage+ ((xtest-MeanImage)*U(:,1:nbVector))*U(:,1:nbVector)';}

_{figure;imagesc(reshape(Reconstruction, 28, 28)'),colormap("gray")}

Question: What is the value of the highest eigenvalue? And how many principal components do we need to keep for reconstruction to get at least 90% of the variance explained?