Correlated noise: How it breaks NMF, and what to do about it

Non-negative matrix factorization (NMF) is an algorithm for decomposing multivariate data into a signal dictionary and its corresponding activations. When applied to experimental data, NMF has to cope with noise, which is often highly correlated. We show that correlated noise can break the Donoho and Stodden separability conditions of a dataset and a regular NMF algorithm will fail to decompose it, even when given freedom to be able to represent the noise as a separate feature. To cope with this issue, we present an algorithm for NMF with a generalized least squares objective function (glsNMF) and derive multiplicative updates for the method together with proving their convergence. The new algorithm successfully recovers the true representation from the noisy data. Robust performance can make glsNMF a valuable tool for analyzing empirical data.