Structured total maximum likelihood: An alternative to structured total least squares

Amir Beck, Yonina C. Eldar

Research output: Contribution to journalArticlepeer-review

Abstract

Linear inverse problems with uncertain measurement matrices appear in many different applications. One of the standard techniques for solving such problems is the total least squares (TLS) method. Recently, an alternative approach has been suggested, based on maximizing an appropriate likelihood function assuming that the measurement matrix consists of random Gaussian variables. We refer to this technique as the total maximum likelihood (TML) method. Here we extend this strategy to the case in which the measurement matrix is structured so that the perturbations are not arbitrary but rather follow a fixed pattern. The resulting estimate is referred to as the structured TML (STML). As we show, the STML can be viewed as a regularized version of the structured TLS (STLS) approach in which the regularization consists of a logarithmic penalty. In contrast to the STLS solution, the STML always exists. Furthermore, its performance in practice tends to be superior to that of the STLS and competitive to other regularized solvers, as we illustrate via several examples. We also consider a few interesting special cases in which the STML can be computed efficiently either by reducing it into a one-dimensional problem regardless of the problem size or by a decomposition via a discrete Fourier transform.

Original languageEnglish
Pages (from-to)2623-2649
Number of pages27
JournalSIAM Journal on Matrix Analysis and Applications
Volume31
Issue number5
DOIs
StatePublished - 2010
Externally publishedYes

Keywords

  • Circulant structures
  • Linear inverse problem
  • Maximum likelihood
  • Nonconvex programming
  • Total least squares

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