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Communication Dans Un Congrès Année : 2018

Tensor and coupled decompositions in block terms: uniqueness and irreducibility

Résumé

Decomposition of a single tensor in a sum of rank-1 terms is a basic and prevalent tool in data analysis. However, in various scenarios, a decomposition in a sum of terms of rank other than one is more faithful to the underlying structure of the data. For this aim, different types of tensor block term decomposition (BTD) have been proposed. When applicable, these models provide advantages such as higher precision and computational efficiency over their rank-1 counterparts. A fundamental concept in decompositions in a sum of terms of rank other than one is (ir)reducibility. In this context, irreducibility means that each term in the sum cannot be further factorized into several distinct terms of smaller rank. Clearly, irreducibility is irrelevant to decompositions in sum of rank-1 terms. Earlier work on the uniqueness of BTD has focused on properties of the factor matrices. In order to avoid reducibility, it was assumed that the block-diagonal terms, also known as core tensors, were generic. Given this assumption, it was proven in earlier work that BTD were generically unique, under mild conditions on the factor matrices. In this talk, we enrich our understanding of block decompositions by presenting new uniqueness results that explicitly take into account non-generic core tensors and (ir)reducibility. Furthermore, we explain how our results provide new insights on CPD and BTD, as well as more general coupled decompositions. For this aim, we formulate an algebraic framework for handling irreducibility. Using this approach, we first prove that BTD (specifically, a type-2 decomposition of a tensor in a sum of rank-$(L_r,M_r,\cdot)$ terms) may be non-unique when the core tensors are irreducible yet not generic. We show that if the diagonal blocks have different size, they cannot cause non-uniqueness; this raises the question whether the set of block dimensions in a BTD can be regarded as a type of diversity. In the second part of this talk, we demonstrate how the more general framework of coupled decompositions sheds new light on irreducibility and on BTD. For this aim, we consider a new type of coupled block decomposition that cannot be represented (in a meaningful way) as a single tensor factorization. First, we show how to extend our analytical framework to characterize its uniqueness, given irreducibility. Then, we present several ways in which this framework subsumes the rank-$(L_r,M_r,\cdot)$ BTD and its uniqueness results. Finally, we show that different block dimensions are not necessarily associated with generic uniqueness.
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Dates et versions

hal-04024260 , version 1 (10-03-2023)

Identifiants

  • HAL Id : hal-04024260 , version 1

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Dana Lahat, Christian Jutten. Tensor and coupled decompositions in block terms: uniqueness and irreducibility. 9th Conference ThRee-way Analysis In Chemistry And Psychology (TRICAP 2018 ), Jun 2018, Angel Fire, New Mexico, United States. ⟨hal-04024260⟩
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