We present a dedicated algorithm for the nonnegative factorization of a correlation matrix from an application in financial engineering. We look for a low-rank approximation. The origin of the problem is discussed in some detail. Next to the description of the algorithm, we prove, by means of a counter example, that an exact nonnegative decomposition of a general positive semidefinite matrix is...

Nonnegative matrix factorization (NMF) is a popular method for multivariate analysis of nonnegative data, the goal of which is decompose a data matrix into a product of two factor matrices with all entries in factor matrices restricted to be nonnegative. NMF was shown to be useful in a task of clustering (especially document clustering). In this paper we present an algorithm for orthogonal nonn...

Nonnegative Matrix Factorization (NMF) is a widely used technique for data representation. Inspired by the expressive power of deep learning, several NMF variants equipped with deep architectures have been proposed. However, these methods mostly use the only nonnegativity while ignoring task-specific features of data. In this paper, we propose a novel deep approximately orthogonal nonnegative m...

A new matrix factorization algorithm which combines two recently proposed nonnegative learning techniques is presented. Our new algorithm, α-PNMF, inherits the advantages of Projective Nonnegative Matrix Factorization (PNMF) for learning a highly orthogonal factor matrix. When the Kullback-Leibler (KL) divergence is generalized to αdivergence, it gives our method more flexibility in approximati...

Abstract. Any given nonnegative matrix A ∈ R can be expressed as the product A = UV for some nonnegative matrices U ∈ R and V ∈ R with k ≤ min{m, n}. The smallest k that makes this factorization possible is called the nonnegative rank of A. Computing the exact nonnegative rank and the corresponding factorization are known to be NP-hard. Even if the nonnegative rank is known a priori, no simple ...

This document summarizes an algorithm for independent low-rank matrix analysis, which was proposed as determined rank-1 multichannel nonnegative matrix factorization in the following published papers: Daichi Kitamura, Nobutaka Ono, Hiroshi Sawada, Hirokazu Kameoka, and Hiroshi Saruwatari, “Efficient multichannel nonnegative matrix factorization exploiting rank-1 spatial model,” Proceedings of I...

Nonnegative Matrix Factorization (NMF) is a data analysis technique which allows compression and interpretation of nonnegative data. NMF became widely studied after the publication of the seminal paper by Lee and Seung (Learning the Parts of Objects by Nonnegative Matrix Factorization, Nature, 1999, vol. 401, pp. 788–791), which introduced an algorithm based on Multiplicative Updates (MU). More...

We give a short combinatorial proof that the nonnegative matrix factorization is an NP-hard problem. Moreover, we prove that NMF remains NP-hard when restricted to 01-matrices, answering a recent question of Moitra. The (exact) nonnegative matrix factorization is the following problem. Given an integer k and a matrix A with nonnegative entries, do there exist k nonnegative rank-one matrices tha...

The notion of low rank approximations arises from many important applications. When the low rank data are further required to comprise nonnegative values only, the approach by nonnegative matrix factorization is particularly appealing. This paper intends to bring about three points. First, the theoretical Kuhn-Tucker optimality condition is described in explicit form. Secondly, a number of nume...

Nonnegative Matrix Factorization (NMF) is a technique to approximate a nonnegative matrix as a product of two smaller nonnegative matrices. The guaranteed nonnegativity of the factors allows interpreting the approximation as an additive combination of features, a distinctive property that other widely used matrix factorization methods do not have. Several advanced methods for computing this fac...