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Concentration of the Frobenius norm of generalized matrix inverses

Ivan Dokmanić 1 Rémi Gribonval 2 
2 PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Modélisation Audio
Inria Rennes – Bretagne Atlantique , IRISA-D5 - SIGNAUX ET IMAGES NUMÉRIQUES, ROBOTIQUE
Abstract : In many applications it is useful to replace the Moore-Penrose pseudoinverse (MPP) by a different 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 by 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 $\ell^p$-minimal general inverses of iid Gaussian matrices, with $1 \leq p \leq 2$. For $p = 1$ we prove exponential concentration of the Frobenius norm of the sparse pseudoinverse; for $p = 2$, we get a similar concentration bound for the MPP. Our proof is based on the convex Gaussian min-max theorem, but unlike previous applications which give asymptotic results, we derive finite-size bounds.
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https://hal.inria.fr/hal-01897046
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Submitted on : Thursday, November 22, 2018 - 7:02:35 PM
Last modification on : Friday, April 8, 2022 - 4:04:02 PM
Long-term archiving on: : Saturday, February 23, 2019 - 1:28:05 PM

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Ivan Dokmanić, Rémi Gribonval. Concentration of the Frobenius norm of generalized matrix inverses. SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, In press, 40 (1), pp.92-121. ⟨10.1137/17M1145409⟩. ⟨hal-01897046v2⟩

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