CUR Matrix Approximation through Convex Optimization for Feature Selection

TL;DR

This paper introduces a new convex optimization-based deterministic CUR matrix approximation algorithm that effectively selects important features and demonstrates its utility in gene expression analysis and protein clustering.

Contribution

The authors develop a novel CUR matrix approximation method with theoretical guarantees, enabling interpretable feature selection and application to biological data analysis.

Findings

The CUR algorithm outperforms SVD in feature interpretability.

The method effectively identifies discriminant proteins in clustering tasks.

Numerical results validate the algorithm's convergence and accuracy.

Abstract

The singular value decomposition (SVD) is commonly used in applications requiring a low rank matrix approximation. However, the singular vectors cannot be interpreted in terms of the original data. For applications requiring this type of interpretation, e.g., selection of important data matrix columns or rows, the approximate CUR matrix factorization can be used. Work on the CUR matrix approximation has generally focused on algorithm development, theoretical guarantees, and applications. In this work, we present a novel deterministic CUR formulation and algorithm with theoretical convergence guarantees. The algorithm utilizes convex optimization, finds important columns and rows separately, and allows the user to control the number of important columns and rows selected from the original data matrix. We present numerical results and demonstrate the effectiveness of our CUR algorithm as a feature selection method on gene expression data. These results are compared to those using the SVD and other CUR algorithms as the feature selection method. Lastly, we present a novel application of CUR as a feature selection method to determine discriminant proteins when clustering protein expression data in a self-organizing map (SOM), and compare the performance of multiple CUR algorithms in this application.