A Symmetric Rank-One Quasi-Newton Method for Nonnegative Matrix Factorization

Shu-Zhen Lai,Hou-Biao Li,Zu-Tao Zhang

doi:10.1155/2014/846483

Abstract

As is well known, the nonnegative matrix factorization (NMF) is a dimension reduction method that has been widely used in image processing, text compressing, signal processing, and so forth. In this paper, an algorithm on nonnegative matrix approximation is proposed. This method is mainly based on a relaxed active set and the quasi-Newton type algorithm, by using the symmetric rank-one and negative curvature direction technologies to approximate the Hessian matrix. The method improves some recent results. In addition, some numerical experiments are presented in the synthetic data, imaging processing, and text clustering. By comparing with the other six nonnegative matrix approximation methods, this method is more robust in almost all cases.

Highlights

An nonnegative matrix factorization (NMF) problem is to decompose a nonnegative matrix V ∈ Rn×m into two nonnegative matrix W ∈ Rn×k and H ∈ Rk×m, such that WH approximate to V as well as possible
A Symmetric Rank-One Quasi-Newton Method for NMF. In the former part, we have discussed the algorithm for nonnegative linear square (NNLS) and in this part we are going to talk about the algorithm for NMF
We present an algorithm for NMF problem, it is different from the famous fnmae method [5] in the following aspects

Summary

A Symmetric Rank-One Quasi-Newton Method for Nonnegative Matrix Factorization

The nonnegative matrix factorization (NMF) is a dimension reduction method that has been widely used in image processing, text compressing, signal processing, and so forth. An algorithm on nonnegative matrix approximation is proposed. This method is mainly based on a relaxed active set and the quasi-Newton type algorithm, by using the symmetric rankone and negative curvature direction technologies to approximate the Hessian matrix. Some numerical experiments are presented in the synthetic data, imaging processing, and text clustering. By comparing with the other six nonnegative matrix approximation methods, this method is more robust in almost all cases

Introduction

A New Method for Nonnegative Least Squares Problems

A Symmetric Rank-One Quasi-Newton Method for NMF

Numerical Experiments

Conclusion