Deep & Reinforcement Learning (CY-802 (B)) - Important Questions

Explain how a linear autoencoder with a single hidden layer trained with mean squared error implements Principal Component Analysis (PCA). Derive how the learned encoder and decoder weight matrices relate to the principal components and eigenvectors of the data covariance matrix. Show the objective and the key steps that lead to the PCA solution.

Discuss the effect of bottleneck size on an autoencoder's ability to reconstruct data and on the quality of learned representations. Compare undercomplete, exactly complete, and overcomplete autoencoders, and explain when each is useful.

Explain the role of regularization techniques in training autoencoders and deep networks: weight decay (L2), dropout, batch normalization, and early stopping. For each technique describe (a) how it modifies the training objective or network behavior, and (b) how it affects generalization and representation learning in autoencoders.

Describe denoising autoencoders. State the training objective used when inputs are stochastically corrupted and explain why denoising training encourages the model to learn robust and useful features compared to a vanilla autoencoder.

Compare and contrast Variational Autoencoders (VAEs) with traditional deterministic autoencoders. Describe the probabilistic generative model, the encoder and decoder roles, and write the Evidence Lower Bound (ELBO) used for training. Include the ELBO formula in your answer.

Explain sparse autoencoders and the use of a sparsity penalty. Write the typical loss function that combines reconstruction mean squared error with a sparsity term using Kullback–Leibler divergence, for example

$$L(\theta)=\frac{1}{N}\sum_{i=1}^N \left\|x^{(i)}-\hat{x}^{(i)}\right\|^2+\beta\sum_{j=1}^h \mathrm{KL}\left(\rho\,\|\,\hat{\rho}_j\right)$$

and explain the meaning of each symbol and how the penalty enforces sparse activations.

Derive the backpropagation gradients and the weight update rules for a single-hidden-layer autoencoder trained with mean squared error. Explicitly show the gradients with respect to the encoder weights and decoder weights and indicate how the chain rule is applied.

Explain batch normalization: give the forward transformation, describe the learned scale and shift parameters, and discuss its effects on training dynamics such as internal covariate shift and learning rate sensitivity. Also explain practical considerations when combining batch normalization with dropout inside autoencoders.

Discuss contractive autoencoders. Write the objective that augments the reconstruction loss with a Jacobian-based contraction penalty and explain how penalizing the Frobenius norm of the encoder Jacobian leads to locally invariant and robust representations.