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

Daniele Gerosa

Research output: ThesisDoctoral Thesis (compilation)

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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 languageEnglish
Supervisors/Advisors
  • Carlsson, Marcus, Supervisor
Award date2022 Apr 26
Publisher
ISBN (Print)978-91-8039-088-0
ISBN (electronic) 978-91-8039-087-3
Publication statusPublished - 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

Free keywords

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

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