Block-Toeplitz/Hankel Structured Total Least Squares

Abstract

Abstract.A structured total least squares problem is considered in which the extended data matrix is partitioned into blocks and each of the blocks is block-Toeplitz/Hankel structured, unstructured, or exact. An equivalent optimization problem is derived and its properties are established. The special structure of the equivalent problem enables us to improve the computational efficiency of the numerical solution methods. By exploiting the structure, the computational complexity of the algorithms (local optimization methods) per iteration is linear in the sample size. Application of the method for system identification and for model reduction is illustrated by simulation examples.

Approximate low-rank factorization with structured factors

“…In this case, we require the approximation X to be nonnegative and in order to achieve this, we impose nonnegativity of the estimate X as an extra constraint in (1). Similarly, in signal processing and system theory the matrix X 0 is Hankel or Toeplitz structured [11] and the relevant modification of (1) is to constrain X to have the same structure. In chemometrics, the measurement errors e i j may have different variances σ 2 v i j , which are known (up to a scaling factor) from the measurement setup or from repeated experiments, see [17,9].…”

“…Where transcription factors down regulate a gene, the elements of C have to be negative to account for this. The constraints (11)(12)(13)(14) in the considered estimation problem (9) (see Section 2) encapsulate this prior knowledge.…”

“…Furthermore, the assumptions (3-7) about the true data matrix X 0 correspond to the constraints (10)(11)(12)(13)(14) in the estimation problem.…”

“…The alternating projections algorithm, see Algorithm 1, is based on the observation that the cost function (9) is quadratic and the constraints (10)(11)(12)(13)(14) are linear in either Algorithm 1 Alternating projections algorithm for solving problem (9-14).…”

“…In Appendix A we describe the implementation of Algorithm 1 for the general case of inhomogeneous weights and constraints (10)(11)(12)(13)(14). Next, we state convergence properties of Algorithm 1.…”