Cancer: Another Algorithm for Subtropical Matrix Factorization

Abstract

Subtropical algebra is a semi-ring over the nonnegative real numbers with standard multiplication and the addition defined as the maximum operator. Factorizing a matrix over the subtropical algebra gives us a representation of the original matrix with element-wise maximum over a collection of nonnegative rank-1 matrices. Such structure can be compared to the well-known Nonnegative Matrix Factorization (NMF) that gives an element-wise sum over a collection of nonnegative rank-1 matrices. Using the maximum instead of sum changes the ‘parts-of-whole’ interpretation of NMF to ‘winner-takes-it-all’ interpretation. We recently introduced an algorithm for subtropical matrix factorization, called Capricorn, that was designed to work on discrete-valued data with discrete noise [Karaev & Miettinen, SDM ’16]. In this paper we present another algorithm, called Cancer, that is designed to work over continuous-valued data with continuous noise – arguably, the more common case. We show that Cancer is capable of finding sparse factors with excellent reconstruction error, being better than either Capricorn, NMF, or SVD in continuous subtropical data. We also show that the winner-takes-it-all interpretation is usable in many real-world scenarios and lets us find structure that is different, and often easier to interpret, than what is found by NMF.