Learning manifold to regularize nonnegative matrix factorization

TL;DR

This paper introduces three methods for learning optimal manifold graphs to improve regularized nonnegative matrix factorization, addressing challenges like graph selection, noise, and nonlinearity in data.

Contribution

It proposes multiple graph learning, adaptive graph learning with feature selection, and multi-kernel graph construction for better NMF regularization.

Findings

Enhanced data representation with learned manifolds

Improved NMF performance on noisy and nonlinear data

Effective graph construction methods for manifold regularization

Abstract

Inthischapterwediscusshowtolearnanoptimalmanifoldpresentationto regularize nonegative matrix factorization (NMF) for data representation problems. NMF,whichtriestorepresentanonnegativedatamatrixasaproductoftwolowrank nonnegative matrices, has been a popular method for data representation due to its ability to explore the latent part-based structure of data. Recent study shows that lots of data distributions have manifold structures, and we should respect the manifold structure when the data are represented. Recently, manifold regularized NMF used a nearest neighbor graph to regulate the learning of factorization parameter matrices and has shown its advantage over traditional NMF methods for data representation problems. However, how to construct an optimal graph to present the manifold prop- erly remains a difficultproblem due to the graph modelselection, noisy features, and nonlinear distributed data. In this chapter, we introduce three effective methods to solve these problems of graph construction for manifold regularized NMF. Multiple graph learning is proposed to solve the problem of graph model selection, adaptive graph learning via feature selection is proposed to solve the problem of constructing a graph from noisy features, while multi-kernel learning-based graph construction is used to solve the problem of learning a graph from nonlinearly distributed data.