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A geometric analysis of convex demixing

Citation

McCoy, Michael Brian (2013) A geometric analysis of convex demixing. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:05202013-091317123

Abstract

Demixing is the task of identifying multiple signals given only their sum and prior information about their structures. Examples of demixing problems include (i) separating a signal that is sparse with respect to one basis from a signal that is sparse with respect to a second basis; (ii) decomposing an observed matrix into low-rank and sparse components; and (iii) identifying a binary codeword with impulsive corruptions. This thesis describes and analyzes a convex optimization framework for solving an array of demixing problems.

Our framework includes a random orientation model for the constituent signals that ensures the structures are incoherent. This work introduces a summary parameter, the statistical dimension, that reflects the intrinsic complexity of a signal. The main result indicates that the difficulty of demixing under this random model depends only on the total complexity of the constituent signals involved: demixing succeeds with high probability when the sum of the complexities is less than the ambient dimension; otherwise, it fails with high probability.

The fact that a phase transition between success and failure occurs in demixing is a consequence of a new inequality in conic integral geometry. Roughly speaking, this inequality asserts that a convex cone behaves like a subspace whose dimension is equal to the statistical dimension of the cone. When combined with a geometric optimality condition for demixing, this inequality provides precise quantitative information about the phase transition, including the location and width of the transition region.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:optimization, demixing, linear inverse problems, geometry, compressed sensing
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Tropp, Joel A.
Thesis Committee:
  • Tropp, Joel A. (chair)
  • Hassibi, Babak
  • Chandrasekaran, Venkat
  • Schulman, Leonard J.
Defense Date:13 May 2013
Funders:
Funding AgencyGrant Number
ONRN00014-08-1-0883
ONRN00014-11-1002
AFOSRFA9550-09-1-0643
Sloan Research FellowshipUNSPECIFIED
Record Number:CaltechTHESIS:05202013-091317123
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:05202013-091317123
ORCID:
AuthorORCID
McCoy, Michael Brian0000-0002-9479-2090
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7726
Collection:CaltechTHESIS
Deposited By: Michael McCoy
Deposited On:21 May 2013 18:26
Last Modified:10 Mar 2014 21:31

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