Convex and Semi-Nonnegative Matrix Factorizations

Abstract

We present several new variations on the theme of nonnegative matrix factorization (NMF). Considering factorizations of the form X = F G T , we focus on algorithms in which G is restricted to contain nonnegative entries, but allow the data matrix X to have mixed signs, thus extending the applicable range of NMF methods. We also consider algorithms in which the basis vectors of F are constrained to be convex combinations of the data points. This is used for a kernel extension of NMF. We provide algorithms for computing these new factorizations and we provide supporting theoretical analysis. We also analyze the relationships between our algorithms and clustering algorithms, and consider the implications for sparseness of solutions. Finally, we present experimental results that explore the properties of these new methods.

Simplicial nonnegative matrix factorization

“…The results are highly competitive with other methods. For Euclidean distance, although our algorithm's sparsity is only less than cNMF [18] (Figure 2), it has lower information loss and higher performance in classification. In addition, especially for KL-divergence, our approach retains the best sparse solutions (Figure 3), while it still has the best result for the other measures.…”

“…More particularly, our algorithm for Euclidean distance is compared to NMF [3], spNMF [17], oNMF [8], and cNMF [18], while the other one with KL-divergence is compared to kl-NMF [3], local non-negative matrix factorization (locNMF) [5], convolutional NMF(conNMF) [19], and Nonsmooth Nonnegative Matrix Factorization (nsNMF) [7]. The implemented codes are at http://www.ee.columbia.edu/ grindlay/code.html.…”

Pattern Recognition Analysis of MR Spectra

“…23 Convex non-negative matrix factorization (convex-NMF) 24 is a variant of NMF that imposes a restriction over the source matrix S to be a convex combination of the input data vectors. This restriction significantly improves the quality of data representation of S. Unlike standard NMF, convex-NMF applies to both non-negative and mixed-sign data matrices.…”

Learning Deep Dictionary for Hyperspectral Image Denoising

“…Since the focus of this paper is validating the advantages of the deep dictionary learning framework, for simplicity, the dictionaries are learned by the convex semi-nonnegative matrix factorization [6], where the decomposition coefficient C i contains only non-negative elements. Noting the role of C i in representing the probability(abundance) that each voxel belongs to certain material types(end-member), this constraint is reasonable.…”