Identifiability of Nonnegative Tucker Decompositions -- Part I: Theory
This work addresses a foundational issue in tensor analysis for data science applications, providing theoretical guarantees for recovering groundtruth sources, though it is incremental by extending known results from nonnegative matrix factorization.
The paper tackles the problem of non-uniqueness in nonnegative Tucker decompositions (nTD) by establishing identifiability conditions, requiring sparsity in matrix factors and full column rank in the core tensor, and proposes procedures to achieve identifiable nTDs through volume minimization.
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.