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

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 ℓ
₁
-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).