Journal
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
Volume 33, Issue 4, Pages 1272-1299Publisher
SIAM PUBLICATIONS
DOI: 10.1137/110859063
Keywords
nonnegative tensor factorization; nonnegative CANDECOMP-PARAFAC; Poisson tensor factorization; Lee-Seung multiplicative updates; majorization-minimization algorithms
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Funding
- U.S. Department of Energy Computational Science Graduate Fellowship [DE-FG02-97ER25308]
- U.S. Department of Energy
- U.S. Department of Energy's National Nuclear Security Administration [DE-AC04-94AL85000]
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Tensors have found application in a variety of fields, ranging from chemometrics to signal processing and beyond. In this paper, we consider the problem of multilinear modeling of sparse count data. Our goal is to develop a descriptive tensor factorization model of such data, along with appropriate algorithms and theory. To do so, we propose that the random variation is best described via a Poisson distribution, which better describes the zeros observed in the data as compared to the typical assumption of a Gaussian distribution. Under a Poisson assumption, we fit a model to observed data using the negative log-likelihood score. We present a new algorithm for Poisson tensor factorization called CANDECOMP-PARAFAC alternating Poisson regression (CP-APR) that is based on a majorization-minimization approach. It can be shown that CP-APR is a generalization of the Lee-Seung multiplicative updates. We show how to prevent the algorithm from converging to non-KKT points and prove convergence of CP-APR under mild conditions. We also explain how to implement CP-APR for large-scale sparse tensors and present results on several data sets, both real and simulated.
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