On uniqueness and computation of the decomposition of a tensor into\n multilinear rank-(1, L_r,L_r) terms

2018/08/07 by Ignat Domanov, Domanov, Ignat, Lieven De Lathauwer +1
Mathematics · Computer Science · #Tensor decomposition and applications #Blind Source Separation Techniques

paper · pdf · doi:10.48550/arxiv.1808.02423

Abstract

Canonical Polyadic Decomposition (CPD) represents a third-order tensor as the\nminimal sum of rank-1 terms. Because of its uniqueness properties the CPD has\nfound many concrete applications in telecommunication, array processing,\nmachine learning, etc. On the other hand, in several applications the rank-1\nconstraint on the terms is too restrictive. A multilinear rank-(M,N,L)\nconstraint (where a rank-1 term is the special case for which M=N=L=1) could\nbe more realistic, while it still yields a decomposition with attractive\nuniqueness properties. In this paper we focus on the decomposition of a tensor\n mathcal T into a sum of multilinear rank-(1,L_r,L_r) terms, r=1,...,R.\nThis particular decomposition type has already found applications in wireless\ncommunication, chemometrics and the blind signal separation of signals that can\nbe modelled as exponential polynomials and rational functions. We find\nconditions on the terms which guarantee that the decomposition is unique and\ncan be computed by means of the eigenvalue decomposition of a matrix even in\nthe cases where none of the factor matrices has full column rank. We consider\nboth the case where the decomposition is exact and the case where the\ndecomposition holds only approximately. We show that in both cases the number\nof the terms R and their "sizes" L₁,...,L_R do not have to be known a\npriori and can be estimated as well. The conditions for uniqueness are easy to\nverify, especially for terms that can be considered "generic". In particular,\nwe obtain the following two generalizations of a well known result on generic\nuniqueness of the CPD (i.e., the case L₁=...=L_R=1): we show that the\nmultilinear rank-(1,L_r,L_r) decomposition of an I\× J\× K tensor\nis generically unique if i) L₁=...=L_R=:L and R\≤\min((J-L)(K-L),I) or\nif ii) \∑ L_R\≤\min((I-1)(J-1),K) and J\≥\max(L_i+L_j).\n

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