This session extends the basic theory and algorithms described in sessions 1-3. We will introduce non-convex approaches to sparse and low-rank recovery, based on sparse Bayesian learning. These algorithms are computationally more intensive than convex relaxations, but empirically can handle many challenging problems in which the convex relaxations fail -- in particular, problems with ``coherent dictionaries''. We will discuss how to approach low-dimensional structure recovery from a Bayesian viewpoint, and what can be said theoretically about the performance of these algorithms. We demonstrate their advantages on problems in image deblurring, neural spike sensing, and matrix estimation. Joint talk by Yi Ma and John Wright.
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