Concentration of the Frobenius norms of pseudoinverses

10/18/2018

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by   Ivan Dokmanić, et al.

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In many applications it is useful to replace the Moore-Penrose pseudoinverse (MPP) by another generalized inverse with more favorable properties. We may want, for example, to have many zero entries, but without giving up too much of the stability of the MPP. One way to quantify stability is to look at how much the Frobenius norm of a generalized inverse exceeds that of the MPP. In this paper we derive finite-size concentration bounds for the Frobenius norm of p-minimal general inverses of iid Gaussian matrices, with 1 < p < 2. For p = 1 we prove exponential concentration of the Frobenius norm of the sparse pseudoinverse; for p = 2, we get similar results for the MPP. Our proof is based on the convex Gaussian min-max theorem, but unlike previous applications we give finite-size concentration results which required development of new techniques.