LIDS & Stats Tea Talk

"A new family of Poisson non-negative matrix factorization models using the shifted log link function"

Non-negative matrix factorization (NMF) is a widely used method to find interpretable, parts-based decompositions of count data. Despite its wide use, little attention has been paid to subtle but important differences between widely used NMF models. In particular, some NMF models assume that the relationship between the latent loadings / factors and the data is additive, whereas other models assume this relationship to be primarily multiplicative. Here, we show that this difference can greatly affect the underlying representations of NMF models and their downstream interpretation. In addition, we introduce a novel Poisson non-negative matrix factorization model that can be tuned to represent either additive OR multiplicative effects. This model formalizes some heuristic approaches currently applied to single cell and text data, and allows us to prove connections between these approaches. Finally, we develop an efficient computational approximation that massively accelerates the fitting of our NMF model in the presence of sparse data.