Spring 2013, EE 520: Special Topics class on Sparse Recovery and Matrix Completion

This will be a special topics course in which we will discuss recent work on (i) sparse recovery / compressive sensing (ii) low-rank matrix completion, (iii) robust low-rank matrix completion (also referred to as robust PCA), and (iv) their applications.  In the first month, we will a review the background needed to understand these papers: (i) linear algebra, (ii) convex optimization and (iii) probability. The rest of the semester will involve a discussion of the key papers on these topics.


Matrix completion solves the following problem: given a matrix with some missing or corrupted entries, how do I recover the original uncorrupted matrix if I know that it is low-rank? In matrix completion, the location of the corrupted entries is known, whereas in robust matrix completion or robust PCA, the corrupted locations are also unknown. The corruptions (outliers) can be very large in magnitude, but occur in only a few rows or columns and thus can be modeled using a sparse matrix.


Sparse Recovery (Compressive Sensing) answers the following question:  how and when can I reconstruct a signal using fewer measurements than the signal length, but using the fact that the signal is sparse or approximately sparse in some domain.


Tentative List of Topics/Papers (suggestions from students are welcome)

Partial list of papers that can be presented

·         Greedy algorithms:

o   Subspace Pursuit paper

o   CoSaMP paper

·         Sparse recovery in sparse noise (outlier)

o   J. Wright and Y. Ma, “Dense error correction via l1-minimization”, IEEE Trans. on Info. Th., vol. 56, no. 7, pp. 3540–3560, 2010.

·         Animesh Biswas

·         Matrix Completion, greedy algorithm

o   K. Lee and Y. Bresler, "ADMiRA: Atomic Decomposition for Minimum Rank Approximation", IEEE Transactions on Information Theory, vol. 56, No. 9, Sep. 2010.

·         Robust Matrix Completion / Robust PCA

o   V. Chandrasekaran, S. Sanghavi, P. A. Parrilo, and A. S. Willsky, “Rank-sparsity incoherence for matrix decomposition”, SIAM Journal on Optimization, vol. 21, 2011.

·         Brian Lois

o   H. Xu, C. Caramanis, and S. Sanghavi, “Robust pca via outlier pursuit”, IEEE Tran. on Information Theorey, vol. 58, no. 5, 2012.38

·         Kevin  Palmowski      

o   M. McCoy and J. Tropp, “Two proposals for robust pca using semidefinite programming”, arXiv:1012.1086v3, 2010.

·         Nicole Kingsley

·         J. A. Tropp, “User-friendly tail bounds for sums of random matrices”, Foundations of Computational Mathematics, vol. 12, no. 4, 2012