Inertial iterative thresholding with applications to sparse and low-rank signal recovery
Johnstone, Patrick
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https://hdl.handle.net/2142/50628
Description
Title
Inertial iterative thresholding with applications to sparse and low-rank signal recovery
Author(s)
Johnstone, Patrick
Issue Date
2014-09-16
Director of Research (if dissertation) or Advisor (if thesis)
Moulin, Pierre
Department of Study
Electrical & Computer Eng
Discipline
Electrical & Computer Engr
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
M.S.
Degree Level
Thesis
Keyword(s)
Iterative shrinkage and thresholding
gradient descent with momentum
the heavy-ball method
the conjugate gradient method
Abstract
This thesis is concerned with a class of methods known collectively as iterative thresholding algorithms. These methods have been used by researchers for several decades to solve various optimization problems that arise in signal processing, inverse problems, pattern recognition and other related fields. One such problem of great interest is compressed sensing, where the goal is to recover a signal that is known to be sparse from fewer linear measurements than the dimension of the signal. Another is low-rank matrix completion
where one wants to recover a low-rank matrix from a subset of revealed entries. A third example is robust principle component analysis (RPCA) where one is given a data matrix and would like to decompose it into a low-rank component and a sparse component. Other examples include total variation denoising and deblurring, and L`1-regularized regression.
Iterative thresholding methods have low complexity, but they typically
take many iterations to converge, especially on ill-conditioned problems. In this thesis we explore how inertia can be used to accelerate iterative thresholding
algorithms. A second problem with iterative thresholding algorithms
is they tend to become trapped in undesirable local minima when the problem is non-convex. We discuss how inertia can help iterative thresholding methods to avoid local minima and propose several schemes to solve well-known non-convex problems.
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