TY - JOUR

T1 - Structured total maximum likelihood

T2 - An alternative to structured total least squares

AU - Beck, Amir

AU - Eldar, Yonina C.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - Circulant structures

KW - Linear inverse problem

KW - Maximum likelihood

KW - Nonconvex programming

KW - Total least squares

UR - http://www.scopus.com/inward/record.url?scp=79251476228&partnerID=8YFLogxK

U2 - 10.1137/090756338

DO - 10.1137/090756338

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AN - SCOPUS:79251476228

SN - 0895-4798

VL - 31

SP - 2623

EP - 2649

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 5

ER -