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

Gerosa, Daniele LU (2022)
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... (More)
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. (Less)
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author
supervisor
opponent
  • Associate Professor Skovgaard Andersen, Martin, Department of Applied Mathematics and Computer Science, Technical University of Denmark.
organization
publishing date
type
Thesis
publication status
published
subject
keywords
compressed sensing, Low-rank Approximation, phase retrieval, Non-convex optimization
pages
227 pages
publisher
Lund University (Media-Tryck)
defense location
Matematikcentrum, Lunds universitet, Sölvegtan 18, Lund. Join via zoom: https://lu-se.zoom.us/j/63651932155
defense date
2022-04-26 15:00:00
ISBN
978-91-8039-088-0
978-91-8039-087-3
language
English
LU publication?
yes
id
b8c73b38-76c2-45f0-a394-f58977b3bd1f
date added to LUP
2022-03-16 13:34:05
date last changed
2022-04-07 07:46:32
@phdthesis{b8c73b38-76c2-45f0-a394-f58977b3bd1f,
  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.}},
  author       = {{Gerosa, Daniele}},
  isbn         = {{978-91-8039-088-0}},
  keywords     = {{compressed sensing; Low-rank Approximation; phase retrieval; Non-convex optimization}},
  language     = {{eng}},
  publisher    = {{Lund University (Media-Tryck)}},
  school       = {{Lund University}},
  title        = {{Non-Convex Methods for Compressed Sensing and Low-Rank Matrix Problems}},
  url          = {{https://lup.lub.lu.se/search/files/115368391/Thesis_Daniele_Gerosa_kappa.pdf}},
  year         = {{2022}},
}