The Total Least Squares Problem in AX≈B: A New Classification with the Relationship to the Classical Works

Abstract

This paper revisits the analysis of the total least squares (TLS) problem AX ≈ B with multiple right-hand sides given by Van Huffel and Vandewalle in the monograph, The Total Least Squares Problem: Computational Aspects and Analysis, SIAM, Philadelphia, 1991. The newly proposed classification is based on properties of the singular value decomposition of the extended matrix ½BjA. It aims at identifying the cases when a TLS solution does or does not exist and when the output computed by the classical TLS algorithm, given by Van Huffel and Vandewalle, is actually a TLS solution. The presented results on existence and uniqueness of the TLS solution reveal subtleties that were not captured in the known literature.

Modification of TLS algorithm for solving ℱ2 linear data fitting problems

“…Let σ n+1 be the kth largest singular value with the multiplicity m k = q + e so that σ n−q > σ n−q+1 = · · · = σ n+1 = · · · = σ n+e > σ n+e+1 . According to [3], consider the following notation of sub-matrices and sub-columns of V ∈ R (n+d)×(n+d) ,…”

“…, v 1,n+e ] is the sub-block corresponding to σ n+1 . Analysis in [3] divides problems (1) into several classes based on the properties of the blocks in (3). If rank([V 12 , V 13 ]) = d, then (1) belongs to the set F (corresponding to generic problems in [6]).…”

“…rank([V 12 , V 13 ]) = d ∧ rank(V 13 ) = d − e. The minimum Frobenius and 2-norm TLS solution of F 1 -problem takes the well-know closed-formwhere † denotes the Moore-Penrose pseudoinverse. However, this is not true for the F 2 -problems; see [3]. Note that for problems in F 3 and S the TLS solution does not exist.…”

“…where † denotes the Moore-Penrose pseudoinverse. However, this is not true for the F 2 -problems; see [3]. Note that for problems in F 3 and S the TLS solution does not exist.…”

The Core Problem within a Linear Approximation Problem $AX\approx B$ with Multiple Right-Hand Sides

“…This book also extends the TLS theory to problems with multiple right-hand sides, i.e., for d > 1. The existence and uniqueness of a TLS solution with d > 1 is then discussed in full generality in the recent paper [10], giving a new classification of all possible cases.…”