Low-Rank Matrix Factorizations with Volume-based Constraints and Regularizations

TL;DR

This paper introduces novel volume-based constraints and regularizations for low-rank matrix factorizations to improve interpretability and uniqueness, with applications in blind source separation and data imputation.

Contribution

It proposes new volume-constrained and volume-regularized low-rank matrix factorization models that enhance interpretability and ensure uniqueness through scatteredness conditions.

Findings

New volume-constrained LRMFs for bounded data and convex polytopes.

Volume-regularized LRMFs promoting clustering and sparsity.

Algorithms enabling practical application of the proposed models.

Abstract

Low-rank matrix factorizations are a class of linear models widely used in various fields such as machine learning, signal processing, and data analysis. These models approximate a matrix as the product of two smaller matrices, where the left matrix captures latent features while the right matrix linearly decomposes the data based on these features. There are many ways to define what makes a component "important." Standard LRMFs, such as the truncated singular value decomposition, focus on minimizing the distance between the original matrix and its low-rank approximation. In this thesis, the notion of "importance" is closely linked to interpretability and uniqueness, which are key to obtaining reliable and meaningful results. This thesis thus focuses on volume-based constraints and regularizations designed to enhance interpretability and uniqueness. We first introduce two new volume-constrained LRMFs designed to enhance these properties. The first assumes that data points are naturally bounded (e.g., movie ratings between 1 and 5 stars) and can be explained by convex combinations of features within the same bounds, allowing them to be interpreted in the same way as the data. The second model is more general, constraining the factors to belong to convex polytopes. Then, two variants of volume-regularized LRMFs are proposed. The first minimizes the volume of the latent features, encouraging them to cluster closely together, while the second maximizes the volume of the decompositions, promoting sparse representations. Across all these models, uniqueness is achieved under the core principle that the factors must be "sufficiently scattered" within their respective feasible sets. Motivated by applications such as blind source separation and missing data imputation, this thesis also proposes efficient algorithms that make these models practical for real-world applications.