Nonnegative Matrix Factorization via Rank-One Downdate

TL;DR

This paper introduces the rank-one downdate algorithm for nonnegative matrix factorization, which adaptively extracts features and demonstrates effectiveness in classifying and analyzing real-world datasets.

Contribution

The paper presents a novel R1D algorithm for NMF that is inspired by SVD and adaptively identifies features, with theoretical and experimental validation.

Findings

R1D effectively classifies articles in nearly separable datasets.

The algorithm successfully identifies features in realistic datasets.

Maximizing the proposed objective aligns with correct classification.

Abstract

Nonnegative matrix factorization (NMF) was popularized as a tool for data mining by Lee and Seung in 1999. NMF attempts to approximate a matrix with nonnegative entries by a product of two low-rank matrices, also with nonnegative entries. We propose an algorithm called rank-one downdate (R1D) for computing a NMF that is partly motivated by singular value decomposition. This algorithm computes the dominant singular values and vectors of adaptively determined submatrices of a matrix. On each iteration, R1D extracts a rank-one submatrix from the dataset according to an objective function. We establish a theoretical result that maximizing this objective function corresponds to correctly classifying articles in a nearly separable corpus. We also provide computational experiments showing the success of this method in identifying features in realistic datasets.