Coseparable Nonnegative Tensor Factorization With T-CUR Decomposition

TL;DR

This paper extends the concept of coseparable nonnegative matrix factorization to tensors, introducing a tensor-based approach that preserves multi-dimensional data correlations and improves efficiency over traditional methods.

Contribution

It proposes a novel coseparable nonnegative tensor factorization method using t-CUR decomposition and randomized index selection, enhancing data representation and computational efficiency.

Findings

Demonstrates improved efficiency over coseparable NMF

Validates t-CUR sampling theory on synthetic and facial datasets

Provides an alternative randomized index selection method for tensor factorization

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.