Non-Negative Matrix Factorization

Across fields such as machine learning, social science, and geology, considerable attention has been given to models that factorize a nonnegative matrix into the product of two or three matrices, subject to nonnegative or row-sum-to-1 constraints. Although these models are to a large extent similar or even equivalent, they are presented under different names, and their similarity is not well known. This paper highlights similarities among five models, latent budget analysis (LBA) and latent class analysis (LCA) from social science, end-member analysis (EMA) from geology, probabilistic latent semantic analysis (PLSA) and nonnegative matrix factorization (NMF) from machine learning. We focus on the identifiability of these models. We prove that the solution of LBA, EMA, LCA, PLSA is unique if and only if the solution of NMF is unique. Consequently, existing uniqueness theorems for NMF directly apply to LBA, EMA, LCA, PLSA, and vice versa. We also provide a brief review of algorithms for the estimation of these models. We illustrate NMF on a sedimentary grain-size distribution dataset from sedimentary geology, and end the paper with a discussion of closely related model: archetypal analysis.

Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for nonnegative data, with applications such as hyperspectral unmixing and topic modeling. NMF is a difficult problem in general (NP-hard), and its solutions are typically not unique. To address these two issues, additional constraints or assumptions are often used. In particular, separability assumes that the basis vectors in the NMF are equal to some columns of the input matrix. In that case, the problem is referred to as separable NMF (SNMF) and can be solved in polynomial-time with robustness guarantees, while identifying a unique solution. However, in real-world scenarios, due to noise or variability, multiple data points may lie near the basis vectors, which SNMF does not leverage. In this work, we rely on the smooth separability assumption, which assumes that each basis vector is close to multiple data points. We explore the properties of the corresponding problem, referred to as smooth SNMF (SSNMF), and examine how it relates to SNMF and orthogonal NMF. We then propose a convex model for SSNMF and show that it provably recovers the sought-after factors, even in the presence of noise. We finally adapt an existing fast gradient method to solve this convex model for SSNMF, and show that it compares favorably with state-of-the-art methods on both synthetic and hyperspectral datasets.

Factor analysis, often regarded as a Bayesian variant of matrix factorization, offers superior capabilities in capturing uncertainty, modeling complex dependencies, and ensuring robustness. As the deep learning era arrives, factor analysis is receiving less and less attention due to their limited expressive ability. On the contrary, contrastive learning has emerged as a potent technique with demonstrated efficacy in unsupervised representational learning. While the two methods are different paradigms, recent theoretical analysis has revealed the mathematical equivalence between contrastive learning and matrix factorization, providing a potential possibility for factor analysis combined with contrastive learning. Motivated by the interconnectedness of contrastive learning, matrix factorization, and factor analysis, this paper introduces a novel Contrastive Factor Analysis framework, aiming to leverage factor analysis's advantageous properties within the realm of contrastive learning. To further leverage the interpretability properties of non-negative factor analysis, which can learn disentangled representations, contrastive factor analysis is extended to a non-negative version. Finally, extensive experimental validation showcases the efficacy of the proposed contrastive (non-negative) factor analysis methodology across multiple key properties, including expressiveness, robustness, interpretability, and accurate uncertainty estimation.

Psychiatry research seeks to understand the manifestations of psychopathology in behavior, as measured in questionnaire data, by identifying a small number of latent factors that explain them. While factor analysis is the canonical tool for this purpose, the resulting factors may not be interpretable, and may also be subject to confounding variables. Moreover, missing data are common, and explicit imputation is often required. To overcome these limitations, we introduce Interpretability Constrained Questionnaire Factorization (ICQF), a non-negative matrix factorization method with regularization tailored for questionnaire data. Our method aims to promote factor interpretability and solution stability. We provide an optimization procedure with theoretical convergence guarantees, and an automated procedure to determine latent dimensionality accurately. We validate these procedures using realistic synthetic data. We demonstrate the effectiveness of our method in a widely used general-purpose questionnaire, in two independent datasets (the Healthy Brain Network and Adolescent Brain Cognitive Development studies). Specifically, we show that ICQF preserves diagnostic information across a range of disorders, outperforming competing methods for smaller dataset sizes, and improves interpretability, as assessed by our clinical research collaborators and co-authors. This suggests that the regularization in our method matches domain characteristics, in addition to satisfying qualitative desiderata.