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On the convex geometry of blind deconvolution and matrix completion


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Krahmer, Felix ; Stöger, Dominik:
On the convex geometry of blind deconvolution and matrix completion.
In: Communications on pure and applied mathematics. 74 (2020) 4. - S. 790-832.
ISSN 0010-3640 ; 1097-0312


Open Access
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Low-rank matrix recovery from structured measurements has been a topic of intense study in the last decade and many important problems like matrix completion and blind deconvolution have been formulated in this framework. An important benchmark method to solve these problems is to minimize the nuclear norm, a convex proxy for the rank. A common approach to establish recovery guarantees for this convex program relies on the construction of a so-called approximate dual certificate. However, this approach provides only limited insight into various respects. Most prominently, the noise bounds exhibit seemingly suboptimal dimension factors. In this paper we take a novel, more geometric viewpoint to analyze both the matrix completion and the blind deconvolution scenario. We find that for both these applications the dimension factors in the noise bounds are not an artifact of the proof, but the problems are intrinsically badly conditioned. We show, however, that bad conditioning only arises for very small noise levels: Under mild assumptions that include many realistic noise levels we derive near-optimal error estimates for blind deconvolution under adversarial noise.

Weitere Angaben

Sprache des Eintrags:Englisch
Institutionen der Universität:Mathematisch-Geographische Fakultät > Mathematik > Juniorprofessur für Data Science
DOI / URN / ID:10.1002/cpa.21957
Open Access: Freie Zugänglichkeit des Volltexts?:Ja
Verlag:Interscience Publ.
Die Zeitschrift ist nachgewiesen in:
Titel an der KU entstanden:Nein
Eingestellt am: 02. Dez 2021 12:45
Letzte Änderung: 03. Jan 2022 09:24
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