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Theory Seminar: Expander codes, Euclidean Sections, and Compressed Sensing
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Venkatesan Guruswami, (Washington University)
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Sunday, 15.6.2008, 13:30
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Room 337-8 Taub Bld.
Classical results in high-dimensional geometry from the 1970's state that a random subspace of R^N of dimension N/2 has "distortion" O(1) w.h.p where the distortion is measured by sqrt{N} times the largest ratio of the L_2 to L_1 norms of a vector in the subspace. (So each vector in the subspace has its mass spread nearly evenly over a constant fraction of the coordinates.) This result is a particular example of the use of the probabilistic method, a technique which is ubiquitous in asymptotic convex geometry. Similar randomized geometric constructions arise in areas like dimensionality reduction and compressed sensing, but it seems that such objects are very hard to come by explicitly.

We will describe constructions inspired by expander codes that can be used to produce subspaces with distortion nearly as good as a random subspace either explicitly or with sublinear randomness. Specifically, we construct an explicit subspace of R^N with dimension N(1-o(1)) and distortion (slightly worse than) poly(log N). The construction combines various ingredients such as Ramanujan graphs, sum-product theorems in finite fields, and Kerdock codes/mutually unbiased bases. We are also able to construct subspaces of R^N of proportional dimension with O(1) distortion using N^delta random bits for any constant delta > 0 (and hence deterministically in sub-exponential time). The talk will also briefly discuss connections to sparse signal recovery (i.e., compressed sensing).

Based on joint works with James Lee, Alexander Razborov, and Avi Wigderson.
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