Factorization Approach to Structured Low-Rank Approximation with Applications

Abstract

We consider the problem of approximating an affinely structured matrix, for example a Hankel matrix, by a low-rank matrix with the same structure. This problem occurs in system identification, signal processing and computer algebra, among others. We impose the low-rank by modeling the approximation as a product of two factors with reduced dimension. The structure of the low-rank model is enforced by introducing a penalty term in the objective function. The proposed local optimization algorithm is able to solve the weighted structured low-rank approximation problem, as well as to deal with the cases of missing or fixed elements. In contrast to approaches based on kernel representations (in linear algebraic sense), the proposed algorithm is designed to address the case of small targeted rank. We compare it to existing approaches on numerical examples of system identification, approximate greatest common divisor problem, and symmetric tensor decomposition and demonstrate its consistently good performance.

Software package for mosaic-Hankel structured low-rank approximation

“…The method opt.solver = 'r' is an implementation of the method in Ishteva et al (2014). (SLRA) is reformulated as…”

“…This paper presents a local optimization-based software package for mosaic Hankel structured low-rank approximation, hosted at https://github.com/slra/slra/ The SLRA package is built upon an implementation of the following methods for structured low-rank approximation: (i) variable projection (kernel representation) Markovsky (2014); Usevich (2014, 2013); Usevich and Markovsky (2014b) and (ii) penalisation (image representation) Ishteva et al (2014). In the variable projection approach, the problem is recast as optimization on a Grassmann manifold Usevich and Markovsky (2014a).…”

“…The previous publication Markovsky and Usevich (2014) is outdated, as it takes into account only the methods of Usevich (2014, 2013); Usevich and Markovsky (2014b)). Here, we describe, in addition, the methods of Usevich and Markovsky (2014b); Ishteva et al (2014); Usevich and Markovsky (2014a); Markovsky (2013); Usevich and Markovsky (2017). Thus, the paper gathers the description of the parameters and options of the methods scattered in the literature.…”

Characterization of Finite Signals with Low-Rank Stft

“…In other words, low-rank approximation problems can be posed and solved in time domain. Hence a rich body of research on Hankel low-rank approximation [10,11,12] may be used. Second, due to orthogonal invariance, nuclear norm minimization [9] can be treated in the time domain.…”

Robust Structured Low-Rank Approximation on the Grassmannian

“…However, as Chu et al [8] and Markovsky [12] state, this solution can be far away from the initialization with no guarantees of finding an actually meaningful approximation to the data. Recently, Ishteva et al [11] have proposed a factorization approach with a cost function that joints the structural and low-rank constraint. The dimension of the two matrix factors is an upper bound on the rank of the approximation and its structure is enforced with a side condition.…”