Derivation of Compressive Sensing Theorems from the Spherical Section property

In this note, we prove several of the main results of compressive sensing from the spherical section property. 1 Compressive sensing In the past four years, there has been extensive activity on compressive sensing. Recently, Kashin and Temlyakov [7] and Zhang [8] have developed simplified proofs of some of the main theorems of compressive sensing using the spherical section property. This notes summarizes proofs of the main theorems from the spherical section inequality. In contrast, Candès and Tao have proved the results using either the uniform uncertainty principle (UUP) or the restricted isometry principle (RIP), while Donoho has used hypotheses called CS1 to CS3. Zhang notes the advantage of the KGG inequality over these other properties, namely, the UUP, RIP, and CS1-3 are not invariant under left-multiplication by an arbitrary nonsingular matrix even though the compressive sensing algorithm is invariant. Definition. Let m,n be two positive integers such that m < n. Let V be an (n − m) dimensional subspace of R. Say that V has the ∆ spherical section property if for any nonzero v ∈ V , ‖v‖1 ‖v‖2 ≥ √ m ∆ . (1) Here, ∆ is called the distortion of V . Zhang quotes Gluskin and Milman [5], who attribute the following theorem to Kashin [6] and Gluskin and Garnaev [4]. Theorem 1. There is a universal constant c0 as follows. Let m,n be two positive integers such that m < n. Let V be an (n−m)-dimensional subspace of R chosen at random with the usual probability measure on choice of subspace. Then with probability at least 1− ec0(n−m), V has the ∆-spherical section property for ∆ = c1(log(n/m) + 1), where c1 is a universal constant. ∗Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada, [email protected]