Identifiability of Nonnegative Tucker Decompositions -- Part I: Theory

TL;DR

This paper investigates the conditions under which nonnegative Tucker decompositions are identifiable, extending matrix factorization results to tensor models, and proposes procedures for achieving uniqueness based on sparsity and rank conditions.

Contribution

It provides new theoretical identifiability results for nonnegative Tucker decompositions by adapting NMF conditions, and introduces procedures to ensure uniqueness using tensor unfoldings or slices.

Findings

Identifiability of nonnegative Tucker decompositions is achievable under sparsity and rank conditions.

Procedures using unfoldings or slices can be employed to obtain identifiable decompositions.

The core tensor's slices or unfoldings with full column rank are sufficient for uniqueness.

Abstract

Tensor decompositions have become a central tool in data science, with applications in areas such as data analysis, signal processing, and machine learning. A key property of many tensor decompositions, such as the canonical polyadic decomposition, is identifiability: the factors are unique, up to trivial scaling and permutation ambiguities. This allows one to recover the groundtruth sources that generated the data. The Tucker decomposition (TD) is a central and widely used tensor decomposition model. However, it is in general not identifiable. In this paper, we study the identifiability of the nonnegative TD (nTD). By adapting and extending identifiability results of nonnegative matrix factorization (NMF), we provide uniqueness results for nTD. Our results require the nonnegative matrix factors to have some degree of sparsity (namely, satisfy the separability condition, or the sufficiently scattered condition), while the core tensor only needs to have some slices (or linear combinations of them) or unfoldings with full column rank (but does not need to be nonnegative). Under such conditions, we derive several procedures, using either unfoldings or slices of the input tensor, to obtain identifiable nTDs by minimizing the volume of unfoldings or slices of the core tensor.