  Non-negative matrix factorization (NMF) is a dimensionality reduction
technique that has shown promise for analyzing noisy data, especially
astronomical data. For these datasets, the observed data may contain negative
values due to noise even when the true underlying physical signal is strictly
positive. Prior NMF work has not treated negative data in a statistically
consistent manner, which becomes problematic for low signal-to-noise data with
many negative values. In this paper we present two algorithms, Shift-NMF and
Nearly-NMF, that can handle both the noisiness of the input data and also any
introduced negativity. Both of these algorithms use the negative data space
without clipping, and correctly recover non-negative signals without any
introduced positive offset that occurs when clipping negative data. We
demonstrate this numerically on both simple and more realistic examples, and
prove that both algorithms have monotonically decreasing update rules.
