Identifiability of Nonnegative Tucker Decompositions -- Part I: Theory

TL;DR

This work addresses the identifiability of nonnegative Tucker decompositions by transferring identifiability concepts from nonnegative matrix factorization to the tensor setting. It develops theory-based identifiability results for order-2 nTDs under separability and the SSC, and extends to order-3 and higher-order nTDs using unfoldings and slices, including min-volume objectives to recover groundtruth cores and factor matrices. The results rely on the SSC, separability, and rank conditions on core slices or unfoldings, yielding essentially unique decompositions in many practically relevant cases, with robustness to noise singled out as a topic for Part II. The findings provide principled guarantees that enable unique groundtruth recovery in structured nonnegative TD models and establish a roadmap for algorithmic development and empirical evaluation in Part II.

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.

Figures (2)

• Figure 1: Comparison of separability (left) and the SSC (right) for the matrix $H$ whose columns lie on the probability simplex, $\Delta^r = \{ x \ | \ x \geq 0, e^\top x = 1 \}$, in the case $r = 3$. On the left, separability requires the columns of $H$ to contain the unit vectors, that is, $H(:,\mathcal{K}) = I_r$ for some $\mathcal{K}$. On the right, the SSC requires $\mathcal{C} \subseteq \mathop{\mathrm{cone}}\nolimits(H)$. Figure adapted from abdolali2021simplex.
• Figure 2: Visualization of $p$-SSC in the case $r=3$ on the plane $\{x\in \mathbb R^r \ | \ e^\top x = 1\}$. On the left, the cones $\mathcal{C}_p$ with $1< p<\sqrt{r-1}$ and $\mathcal{C}_q$ with $\sqrt{r-1}< q<\sqrt r$. On the right, the cone $\mathcal{C}\equiv \mathcal{C}_{\sqrt {r-1}}$ and the nonnegative orthant $\mathbb R^r_+\equiv \mathcal{C}_1$.

Theorems & Definitions (59)

• Definition 2.1: Essential Uniqueness of TD
• Definition 2.2: Slices of an order-$3$ tensor
• Definition 2.3: Unfoldings of an order-$3$ tensor
• Definition 2.4: Kronecker products
• Definition 2.5
• Theorem 2.6
• Definition 2.7
• Theorem 2.8: Identifiability of min-vol NMF
• Remark 2.9: Nonnegativity of $X$ and $W$
• Theorem 3.1