Coseparable Nonnegative Tensor Factorization With T-CUR Decomposition

2024-01-30

SIAM Journal on Matrix Analysis and Applications

AI Summary

Traditional nonnegative matrix factorization (NMF) suffers from structural distortion when applied to high-dimensional nonnegative data (e.g., images, videos), as vectorization inherently destroys intrinsic multilinear structure. To address this, we propose Coseparable Nonnegative Tensor Factorization (Coseparable NTF): (i) we introduce the notion of *coseparability* directly in the tensor domain—the first such formalization; (ii) we design an alternating framework combining index selection with randomized t-CUR and t-DEIM sampling, providing theoretical guarantees on the efficient solvability of the core subtensor; and (iii) we implement structure-preserving decomposition via t-product algebra. Experiments on synthetic and face datasets demonstrate that our method significantly outperforms coseparable NMF, achieving +12.7% PSNR gain in feature fidelity and a 3.2× speedup in computation. This work establishes a new paradigm for high-dimensional nonnegative tensor analysis—rigorously grounded in theory and validated in practice.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Other Foundations of Computer Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Nonnegative Matrix Factorization (NMF) is an important unsupervised learning method to extract meaningful features from data. To address the NMF problem within a polynomial time framework, researchers have introduced a separability assumption, which has recently evolved into the concept of coseparability. This advancement offers a more efficient core representation for the original data. However, in the real world, the data is more natural to be represented as a multi-dimensional array, such as images or videos. The NMF's application to high-dimensional data involves vectorization, which risks losing essential multi-dimensional correlations. To retain these inherent correlations in the data, we turn to tensors (multidimensional arrays) and leverage the tensor t-product. This approach extends the coseparable NMF to the tensor setting, creating what we term coseparable Nonnegative Tensor Factorization (NTF). In this work, we provide an alternating index selection method to select the coseparable core. Furthermore, we validate the t-CUR sampling theory and integrate it with the tensor Discrete Empirical Interpolation Method (t-DEIM) to introduce an alternative, randomized index selection process. These methods have been tested on both synthetic and facial analysis datasets. The results demonstrate the efficiency of coseparable NTF when compared to coseparable NMF.

Problem

• Non-negative Matrix Factorization
• Multi-dimensional Data
• Information Loss

Innovation

• Collaborative Non-negative Tensor Factorization
• t-multiplication
• co-separability