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Preprints, Working Papers, ... Year : 2017

Beyond Moore-Penrose Part II: The Sparse Pseudoinverse

Abstract

This is the second part of a two-paper series on generalized inverses that minimize matrix norms. In Part II we focus on generalized inverses that are minimizers of entrywise p norms whose main representative is the sparse pseudoinverse for $p = 1$. We are motivated by the idea to replace the Moore-Penrose pseudoinverse by a sparser generalized inverse which is in some sense well-behaved. Sparsity implies that it is faster to apply the resulting matrix; well-behavedness would imply that we do not lose much in stability with respect to the least-squares performance of the MPP. We first address questions of uniqueness and non-zero count of (putative) sparse pseu-doinverses. We show that a sparse pseudoinverse is generically unique, and that it indeed reaches optimal sparsity for almost all matrices. We then turn to proving our main stability result: finite-size concentration bounds for the Frobenius norm of p-minimal inverses for $1 ≤ p ≤ 2$. Our proof is based on tools from convex analysis and random matrix theory, in particular the recently developed convex Gaussian min-max theorem. Along the way we prove several results about sparse representations and convex programming that were known folklore, but of which we could find no proof.
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Dates and versions

hal-01547283 , version 1 (30-06-2017)
hal-01547283 , version 2 (13-07-2017)

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Ivan Dokmanić, Rémi Gribonval. Beyond Moore-Penrose Part II: The Sparse Pseudoinverse. 2017. ⟨hal-01547283v2⟩
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