Almost Euclidean subspaces of ℓ N1 VIA expander codes

Abstract

We give an explicit (in particular, deterministic polynomial time) construction of subspaces X⊆ℝN of dimension (1−o(1))N such that for every xX,

$$ (\log N)^{ - O(\log \log \log N)} \sqrt N \left\| x \right\|_2 \leqslant \left\| x \right\|_1 \leqslant \sqrt N \left\| x \right\|_2 $$

. If we are allowed to use N 1/log logNN o(1) random bits and dim(X) ⩾ (1−η)N for any fixed constant η, the lower bound can be further improved to \( (\log N)^{ - O(1)} \sqrt N \left\| x \right\|_2 \).

Through known connections between such Euclidean sections of ℓ1 and compressed sensing matrices, our result also gives explicit compressed sensing matrices for low compression factors for which basis pursuit is guaranteed to recover sparse signals. Our construction makes use of unbalanced bipartite graphs to impose local linear constraints on vectors in the subspace, and our analysis relies on expansion properties of the graph. This is inspired by similar constructions of error-correcting codes.

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Correspondence to Venkatesan Guruswami.

Additional information

A preliminary version of this paper appeared in the Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, January 2008.

This work was done when the author was at Department of Computer Science and Engineering, University of Washington, Seattle, WA, and a member in the School of Mathematics, Institute for Advanced Study, Princeton, NJ. Research supported in part by NSF CCF-0343672, a Packard Fellowship, and NSF grant CCF-0324906 to the IAS.

Research supported in part by NSF CCF-0644037.

Supported by NSF grant ITR-0324906 and by the Russian Foundation for Basic Research.

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Guruswami, V., Lee, J.R. & Razborov, A. Almost Euclidean subspaces of ℓ N1 VIA expander codes. Combinatorica 30, 47–68 (2010). https://doi.org/10.1007/s00493-010-2463-9

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Mathematics Subject Classification (2000)

  • 68R05
  • 68P30
  • 51N20