A Projective Approach to Nonnegative Matrix Factorization
Abstract: Nonnegative matrix factorization as a tool in data science to analyse the structure of the underlying dataset appears in various applications and enjoys great popularity. Consider a given square matrix $A$. The symmetric nonnegative matrix factorization aims for a nonnegative low-rank approximation $A\approx XX^T$ to $A$, where $X$ is entrywise nonnegative and of given order. This setting can be seen as demanding a so-called completely positive approximation of $A$. In this paper we introduce an alternating projection type approach to this setting in order to obtain symmetric nonnegative matrix factorizations. Moreover, considering a general rectangular input matrix $A$, the general nonnegative matrix factorization again aims for a nonnegative low-rank approximation to $A$ which is now of the type $A\approx XY$ for entrywise nonnegative matrices $X,Y$ of given order. Here we introduce a new perspective motivated by our results in the symmetric case in order to derive nonnegative matrix factorizations even in this general setting.
Keywords: nonnegative matrix factorization, symmetric nonnegative matrix factorization, low-rank approximation, completely positive matrices
Category 1: Linear, Cone and Semidefinite Programming
Category 2: Applications -- Science and Engineering
Citation: P. Groetzner, A Projective Approach to Nonnegative Matrix Factorization. Preprint, 2019.
Entry Submitted: 04/23/2019
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