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P1. Image Representation and PCA Sparsity (25 points)

1. What are orthogonal basis functions? How can an input image patch be expressed as a combination of orthogonal basis functions?

2. Give two examples of orthogonal basis functions.

3. Give a method for learning a set of basis vectors given a training set of images. What form do these basis functions take if the image is shift-invariant?

4. How can we represent images in terms of a linear combination of over-complete basis functions by imposing a sparsity constraint?

5. What is the miracle of sparsity? Describe L1 sparsity and show, for a simple example, how it results in a sparse representation.

P2. Dictionaries, Mixtures of Gaussians, Mini-Epitomes, EM (25 points)

1. What is the k-means algorithm? What are the means, the assignment variable, and k? What are its convergence properties? What are the advantages of k-means++?

2. How can k-means be used to learn a set of dictionary elements for image patches?

3. What is a mixture of Gaussian distribution? And how does k-means relate to a mixture of Gaussian distributions?

4. What are mini-epitones? How do they deal with shift-invariance? What algorithm is used to learn them? How well can they represent images?

5. What is the Expectation-Maximization (EM) algorithm? How can EM be applied to learning a mixture of Gaussian distributions? Describe why the EM algorithm converges.