A variable neighborhood search heuristic for nonnegative matrix factorization with application to microarray data

Abstract

Nonnegative matrix factorization (NMF) has become a popular method for establishing a low-dimensional approximation of a two-mode (nonnegative) data matrix and, in some instances, to also establish partitions for the objects associated with the two modes of the matrix. Although similar to singular-value-decomposition, as its name implies, NMF requires nonnegative elements for the factors and this assures a ‘sum of the parts’ fit to the data. There are a variety of alternative objective functions and heuristic methods for NMF. Using both simulated and real-world two-mode data, we demonstrate that a multiple restart (multistart) heuristic for NMF commonly fails to produce optimal objective function values. A new variable neighborhood search (VNS) heuristic is shown to outperform the multistart approach with respect to both solution quality and computation time. Although the objective function value improvements associated with VNS are often small (on a percentage basis), such improvements can sometimes lead to differences in the partitions obtained for the row and column objects. Application to two well-known microarray datasets is used to support the merits of the proposed VNS heuristic.