Identifiability of Nonnegative Tensor Decompositions via Positive Scattering

TL;DR

The paper introduces a new identifiability criterion for nonnegative tensor decompositions using a positive scattering measure that captures geometric constraints from nonnegativity and support structure. It establishes two thresholds—2|S|−2 for minimal rank and 2|S|−1 for uniqueness—based on a splitting inequality that combines dimensionality and connectivity. The method outperforms classical conditions like Kruskal and Lovitz–Petrov, especially in sparse settings where reshaping fails.

Why It Matters

Nonnegative tensor decompositions are widely used in signal processing, neuroscience, and data mining, where interpretability and uniqueness are crucial. This work enables stronger guarantees for recovery in real-world sparse data scenarios where prior methods fail, improving reliability in applications like topic modeling and hyperspectral unmixing.

Key Findings

• A positive scattering term quantifies geometric rigidity from nonnegativity and support constraints, enabling stronger identifiability beyond linear algebraic conditions.
• The proposed criteria strictly extend Kruskal and Lovitz–Petrov conditions, certifying uniqueness in cases where those methods fail even after tensor reshaping.