Identifiability of Nonnegative Tucker Decompositions -- Part I: Theory

2025-05-19

AI Summary

This paper addresses the inherent unidentifiability of nonnegative Tucker decomposition (nTD), establishing— for the first time—theoretical foundations for its uniqueness. Building upon identifiability results from nonnegative matrix factorization (NMF), we propose a joint criterion that combines sparsity-inducing structural assumptions on factor matrices (e.g., separability or sufficient scattering) with low-rank conditions on core tensor slices or unfoldings, thereby overcoming the fundamental lack of uniqueness in conventional Tucker models. Methodologically, we integrate tensor unfolding and slicing analysis, volume-minimization optimization, and rank-based linear algebraic verification to derive multiple verifiable sufficient conditions for uniqueness. We further design constructive algorithms for identifiable nTD, leveraging either unfolding- or slicing-based strategies. Experiments demonstrate that the proposed framework substantially improves the reliability and interpretability of latent factor recovery from high-dimensional nonnegative data.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Study identifiability of nonnegative Tucker decompositions (nTD)
• Extend NMF uniqueness results to nTD with sparsity conditions
• Develop procedures for identifiable nTD via core tensor optimization

Innovation

• Nonnegative Tucker Decomposition with sparsity conditions
• Uniqueness via separability or sufficiently scattered conditions
• Minimizing core tensor volume for identifiable decompositions