Linear systems with a canonical polyadic decomposition constrained solution: Algorithms and applications

Abstract

Summary Real‐life data often exhibit some structure and/or sparsity, allowing one to use parsimonious models for compact representation and approximation. When considering matrix and tensor data, low‐rank models such as the (multilinear) singular value decomposition, canonical polyadic decomposition (CPD), tensor train, and hierarchical Tucker model are very common. The solution of (large‐scale) linear systems is often structured in a similar way, allowing one to use compact matrix and tensor models as well. In this paper, we focus on linear systems with a CPD‐constrained solution (LS‐CPD). Our main contribution is the development of optimization‐based and algebraic methods to solve LS‐CPDs. Furthermore, we propose a condition that guarantees generic uniqueness of the obtained solution. We also show that LS‐CPDs provide a broad framework for the analysis of multilinear systems of equations. The latter are a higher‐order generalization of linear systems, similar to tensor decompositions being a generalization of matrix decompositions. The wide applicability of LS‐CPDs in domains such as classification, multilinear algebra, and signal processing is illustrated.

Augmenting interictal mapping with neurovascular coupling biomarkers by structured factorization of epileptic EEG and fMRI data

“…We constructed a flattened ‘design matrix’ in (13) and obtained a rough estimate for as via regression—albeit this does not yet disentangle and . To obtain initializations for the individual spatial factors, we exploit the fact that in every ROI the Khatri–Rao product of the -th columns of and corresponds to a rank-1 constraint when folded into a matrix ( Beckmann, Smith, 2005 , Boussé, Vervliet, Domanov, Debals, De Lathauwer, 2018 ); hence, a rank-1 truncated singular value decomposition of the folded -th column of leads to the desired vectors ( Beckmann and Smith, 2005 ), which are further refined via a constrained Gauss–Newton algorithm ( Boussé et al., 2018 ). We approximated the residual of the fMRI data under the initialized (coupled) factors using a rank- truncated SVD to capture fMRI nuisances.…”

NLS Algorithm for Kronecker-Structured Linear Systems with a CPD Constrained Solution

“…Even though the decomposition of a tensor that is known explicitly is a prevalent problem in signal processing and machine learning [1], [2], we often want to compute a decomposition of a tensor that is only known via linear measurements [3]. Applications can be found in a wide range of domains such as signal processing [3], [4], system identification [5], [6], pattern recognition [3], [7]- [9], and scientific computing [10]- [13]. By limiting ourselves to a canonical polyadic decomposition (CPD) in this paper, we can formulate the problem as a linear system of equations with a CPD constrained solution (LS-CPD) [3], i.e., Ax = b with x = vec (CPD).…”

“…Applications can be found in a wide range of domains such as signal processing [3], [4], system identification [5], [6], pattern recognition [3], [7]- [9], and scientific computing [10]- [13]. By limiting ourselves to a canonical polyadic decomposition (CPD) in this paper, we can formulate the problem as a linear system of equations with a CPD constrained solution (LS-CPD) [3], i.e., Ax = b with x = vec (CPD). Or, equivalently, we want to compute a CPD of a tensor X = unvec (x) that is only defined implicitly via the solution of a linear system.…”

“…By fully exploiting all structure of the measurement matrix A, the computational complexity of a dedicated algorithm can be significantly reduced, enabling efficient processing for large-scale problems [3]. For example, if A is equal to the identity (or a diagonal) matrix, the problem reduces to a (weighted) CPD of a known tensor, allowing efficient computations [14]- [16].…”

7th Workshop on Matrix Equations and Tensor Techniques

“…Boussé et al consider the solution of tensor‐structured linear systems. More specifically, it is assumed that the solution vector, reshaped as a tensor, is in canonical polyadic decomposition.…”