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Non-negative matrix factorization (NMF) is a dimensionality reductiontechnique that has shown promise for analyzing noisy data, especiallyastronomical data. For these datasets, the observed data may contain negativevalues due to noise even when the true underlying physical signal is strictlypositive. Prior NMF work has not treated negative data in a statisticallyconsistent manner, which becomes problematic for low signal-to-noise data withmany negative values. In this paper we present two algorithms, Shift-NMF andNearly-NMF, that can handle both the noisiness of the input data and also anyintroduced negativity. Both of these algorithms use the negative data spacewithout clipping, and correctly recover non-negative signals without anyintroduced positive offset that occurs when clipping negative data. Wedemonstrate this numerically on both simple and more realistic examples, andprove that both algorithms have monotonically decreasing update rules.

Algorithms for Non-Negative Matrix Factorization on Noisy Data With Negative Values