Kruskal's Permutation Lemma and the Identification of CANDECOMP/PARAFAC and Bilinear Models with Constant Modulus Constraints

Abstract

Abstract-CANDECOMP/PARAFAC (CP) analysis is an extension of low-rank matrix decomposition to higher-way arrays, which are also referred to as tensors. CP extends and unifies several array signal processing tools and has found applications ranging from multidimensional harmonic retrieval and angle-carrier estimation to blind multiuser detection. The uniqueness of CP decomposition is not fully understood yet, despite its theoretical and practical significance. Toward this end, we first revisit Kruskal's Permutation Lemma, which is a cornerstone result in the area, using an accessible basic linear algebra and induction approach. The new proof highlights the nature and limits of the identification process. We then derive two equivalent necessary and sufficient uniqueness conditions for the case where one of the component matrices involved in the decomposition is full column rank. These new conditions explain a curious example provided recently in a previous paper by Sidiropoulos, who showed that Kruskal's condition is in general sufficient but not necessary for uniqueness and that uniqueness depends on the particular joint pattern of zeros in the (possibly pretransformed) component matrices. As another interesting application of the Permutation Lemma, we derive a similar necessary and sufficient condition for unique bilinear factorization under constant modulus (CM) constraints, thus providing an interesting link to (and unification with) CP.

Multi-Way Compressed Sensing for Sparse Low-Rank Tensors

“…We can now prove Theorem 2. Proof: From [26] (see also [14] for a deterministic counterpart), we know that PARAFAC is almost surely identifiable if the loading matrices are randomly drawn from an absolutely continuous distribution with respect to the Lebesgue measure in , is full column rank, and . Full rank of is ensured almost surely by Lemma 1.…”

Blind Identification of Underdetermined Mixtures by Simultaneous Matrix Diagonalization

“…The derivation shows that these deterministic conditions are sufficient for essential uniqueness of the PARAFAC decomposition (4). This result has independently, in an entirely different way, been obtained in [39]. For generic mixtures, the conditions reduce to the dimensionality constraints (15) and (16) [20], [58].…”

“…The following theorem establishes a condition under which essential uniqueness is guaranteed [39], [40], [53], [59].…”

Tensors : A brief introduction

“…However, these conditions are generally only sufficient [41], and often much more restrictive. The most well known is that published by Kruskal [47] and extended later in [73], [81]; alternate proofs have been derived in [68], [49].…”