Non-Convex Methods for Compressed Sensing and Low-Rank Matrix Problems

Daniele Gerosa

Research output: ThesisDoctoral Thesis (compilation)

Abstract

In this thesis we study functionals of the type $$\mathcal{K}_{f,A,\b}(\x)= \mathcal{Q}(f)(\x) + \|A\x - \b \| ^2$$, where $$A$$ is a linear map, $$\b$$ a measurements vector and $$\mathcal{Q}$$ is a functional transform called \emph{quadratic envelope}; this object is a very close relative of the \emph{Lasry-Lions envelope} and its use is meant to regularize the functionals $$f$$. Carlsson and Olsson investigated in earlier works the connections between the functionals $$\mathcal{K}_{f,A,\b}$$ and their unregularized counterparts $$f(\x) + \|A\x - \b \| ^2$$. For certain choices of $$f$$ the penalty $$\mathcal{Q}(f)(\cdot)$$ acts as sparsifying agent and the minimization of $$\mathcal{K}_{f,A,\b}(\x)$$ delivers sparse solutions to the linear system of equations $$A\x = \b$$. We prove existence and uniqueness results of the sparse (or low rank, since the functional $$f$$ can have any Hilbert space as domain) global minimizer of $$\mathcal{K}_{f,A,\b}(\x)$$ for some instances of $$f$$, under Restricted Isometry Property conditions on $$A$$. The theory is complemented with robustness results and a wide range of numerical experiments, both synthetic and from real world problems.
Original language English Carlsson, Marcus, Supervisor 2022 Apr 26 Lund University (Media-Tryck) 978-91-8039-088-0 978-91-8039-087-3 Published - 2022

Bibliographical note

Defence details
Date: 2022-04-26
Time: 15:00
Place: Matematikcentrum, Lunds universitet, Sölvegtan 18, Lund. Join via zoom: https://lu-se.zoom.us/j/63651932155
External reviewer(s)
Name: Skovgaard Andersen, Martin
Title: Associate Professor
Affiliation: Department of Applied Mathematics and Computer Science, Technical University of Denmark.
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Subject classification (UKÄ)

• Mathematical Analysis

Keywords

• compressed sensing
• Low-rank Approximation
• phase retrieval
• Non-convex optimization

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• Bias Versus Non-Convexity in Compressed Sensing

Gerosa, D., Carlsson, M. & Olsson, C., 2022, 64, 4, p. 379-394

Research output: Contribution to journalArticlepeer-review

Open Access
• Relaxations for Non-Separable Cardinality/Rank Penalties

Olsson, C., Gerosa, D. & Carlsson, M., 2021, 2021 IEEE/CVF International Conference on Computer Vision Workshops (ICCVW). IEEE - Institute of Electrical and Electronics Engineers Inc., p. 162-171 10 p. (IEEE International Conference on Computer Vision Workshops).

Research output: Chapter in Book/Report/Conference proceedingPaper in conference proceedingpeer-review

Open Access
• An unbiased approach to compressed sensing

Carlsson, M., Gerosa, D. & Olsson, C., 2020 Nov, 36, 11, 115014.

Research output: Contribution to journalArticlepeer-review

Open Access