Restricted invertibility revisited

Suppose that m, nεN and that A:R^m→Rⁿ is a linear operator. It is shown here that if k, rεN satisfy k<r≤rank(A) then there exists a subset σ ⊇ {1,…, m} with | σ | = k such that the restriction of A to Rσ⊇R^m is invertible, and moreover the operator norm of the inverse A-1:A(Rσ)→R^m is at most a constant multiple of the quantity (formula presented), where s₁(A)≥…≥s_m(A) are the singular values of A. This improves over a series of works, starting from the seminal Bourgain-Tzafriri Restricted Invertibility Principle, through the works of Vershynin, Spielman-Srivastava and Marcus-Spielman-Srivastava. In particular, this directly implies an improved restricted invertibility principle in terms of Schatten-von Neumann norms.