Autoencoder. Autoencoder is a classic type of neural networks for unsupervised learning.

a. Demonstrate that PCA is a special case of the autoencoder by showing that the loss function of PCA is equivalent to the mean-squared error of the autoencoder with linear activation, where the encoder and decoder share the same parameters.

b. Implement an autoencoder with nonlinear activation, and use (1) mean-squared-error and (2) binary cross-entropy as the loss function. Note that when using the binary cross-entropy

loss, each data point should be normalized into $[0,1]$. Compare them with PCA on the MNIST data set, which can be found at http://yann.lecun.com/exdb/mnist/, in terms of classification accuracy and the visualization of reconstructed images. For the classification task, first use the autoencoder or PCA to obtain the features of the data and then train a linear SVM or logistic regression model using the features.