Discuss the challenges and limitations of applying score matching as a pretraining strategy beyond the first hidden layer in your deep neural network architecture. Explain why it may not be effective for deeper layers and propose an alternative approach to overcome these limitations.

Score matching is primarily suited for pretraining shallow models, particularly the first hidden layer. Its inability to work with lower bounds on log ˜p(x) makes it incompatible with deeper layers that have more complex interactions between units. To overcome this limitation, we can use alternative pretraining strategies like contrastive divergence or pseudolikelihood.