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Dimensionality reduction methods for count data are critical to a wide rangeof applications in medical informatics and other fields where modelinterpretability is paramount. For such data, hierarchical Poisson matrixfactorization (HPF) and other sparse probabilistic non-negative matrixfactorization (NMF) methods are considered to be interpretable generativemodels. They consist of sparse transformations for decoding their learnedrepresentations into predictions. However, sparsity in representation decodingdoes not necessarily imply sparsity in the encoding of representations from theoriginal data features. HPF is often incorrectly interpreted in the literatureas if it possesses encoder sparsity. The distinction between decoder sparsityand encoder sparsity is subtle but important. Due to the lack of encodersparsity, HPF does not possess the column-clustering property of classical NMF-- the factor loading matrix does not sufficiently define how each factor isformed from the original features. We address this deficiency byself-consistently enforcing encoder sparsity, using a generalized additivemodel (GAM), thereby allowing one to relate each representation coordinate to asubset of the original data features. In doing so, the method also gains theability to perform feature selection. We demonstrate our method on simulateddata and give an example of how encoder sparsity is of practical use in aconcrete application of representing inpatient comorbidities in Medicarepatients.

Sparse encoding for more-interpretable feature-selecting representations in probabilistic matrix factorization