Matrix Factorization Techniques in Machine Learning, Signal Processing, and Statistics

Abstract

Compressed sensing is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Sparse coding represents a signal as a sparse linear combination of atoms, which are elementary signals derived from a predefined dictionary. Compressed sensing, sparse approximation, and dictionary learning are topics similar to sparse coding. Matrix completion is the process of recovering a data matrix from a subset of its entries, and it extends the principles of compressed sensing and sparse approximation. The nonnegative matrix factorization is a low-rank matrix factorization technique for nonnegative data. All of these low-rank matrix factorization techniques are unsupervised learning techniques, and can be used for data analysis tasks, such as dimension reduction, feature extraction, blind source separation, data compression, and knowledge discovery. In this paper, we survey a few emerging matrix factorization techniques that are receiving wide attention in machine learning, signal processing, and statistics. The treated topics are compressed sensing, dictionary learning, sparse representation, matrix completion and matrix recovery, nonnegative matrix factorization, the Nyström method, and CUR matrix decomposition in the machine learning framework. Some related topics, such as matrix factorization using metaheuristics or neurodynamics, are also introduced. A few topics are suggested for future investigation in this article.

Keywords

• compressed sensing
• dictionary learning
• sparse approximation
• matrix completion
• nonnegative matrix factorization