Expander codes over reals, Euclidean sections, and compressed sensing

  • Authors:
  • Venkatesan Guruswami;James R. Lee;Avi Wigderson

  • Affiliations:
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA;Department of Computer Science and Engineering, University of Washington, Seattle, WA;School of Mathematics, Institute for Advanced Study, Princeton, NJ

  • Venue:
  • Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
  • Year:
  • 2009

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Abstract

Classical results from the 1970's state that w.h.p. a random subspace of N-dimensional Euclidean space of proportional (linear in N) dimension is "well-spread" in the sense that vectors in the subspace have their l2 mass smoothly spread over a linear number of coordinates. Such well-spread subspaces are intimately connected to low distortion embeddings, compressed sensing matrices, and error-correction over reals. We describe a construction inspired by expander/Tanner codes that can be used to produce well-spread subspaces of Ω(N) dimension using sub-linear randomness (or in subexponential time). These results were presented in our paper [10]. We also discuss the connection of our subspaces to compressed sensing, and describe a near-linear time iterative recovery algorithm for compressible signals.