NDLI: Non-asymptotic analysis of compressed sensing random matrices: An U-statistics approach

Content Provider	IEEE Xplore Digital Library
Author	Lim, F. Stojanovic, V.M.
Copyright Year	2012
Description	Author affiliation: Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139 (Lim, F.; Stojanovic, V.M.)
Abstract	We apply Heoffding's U-statistics to obtain non-asymptotic analysis for compressed sensing (CS) random matrices. These powerful (U-statistics) tools appear to apply naturally to CS theory, in particular here we focus on one particular large deviation result. We chose two applications to outline how U-statistics may apply to various CS recovery guarantees. Pros, cons, and further directions of the approach, are discussed. Restricted isometries of random matricies have well-regarded importance in CS. They guarantee i) uniqueness of sparse solutions, and ii) robust recovery. The fraction of size-k submatrices (out of all (n/k) of them), that satisfy CS-type restricted isometeries, is an U-statistic. Concentration of U-statistics predict the “average-case” behavior of such isometries. U-statistics related to Fuchs' conditions for ℓ1-minimization support recovery, are derived. This leads to bounds on the fraction of recoverable k-supports. Empirically, we observe significant improvement over a recent large deviation (non-asymptotic) bound by Donoho & Tanner, for some practical system sizes with large undersampling. The results apply regardless of column distribution, e.g. Gaussian, Bernoulli, etc. Similar concentration behavior has been empirically observed, when the sampling matrix is constructed using pseudorandom sequences (important in practice).
Starting Page	2511
Ending Page	2515
File Size	633218
Page Count	5
File Format	PDF
ISBN	9781457720529
ISSN	15503607
e-ISBN	9781457720536
e-ISBN	9781457720512
DOI	10.1109/ICC.2012.6364237
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2012-06-10
Publisher Place	Canada
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Kernel Compressed sensing Sparse matrices Vectors Robustness Approximation methods Silicon restricted isometries compressed sensing pseudorandom sequences random matrices
Content Type	Text
Resource Type	Article

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