We introduce and demonstrate the variational autoencoder (VAE) for probabilistic non-negative matrix factorisation (PAE-NMF). We design a network which can perform non-negative matrix factorisation (NMF) and add in aspects of a VAE to make the coefficients of the latent space probabilistic. By restricting the weights in the final layer of the network to be non-negative and using the non-negative Weibull distribution we produce a probabilistic form of NMF which allows us to generate new data and find a probability distribution that effectively links the latent and input variables. We demonstrate the effectiveness of PAE-NMF on three heterogeneous datasets: images, financial time series and genomic.

Non-negative matrix factorization (NMF) is a dimensionality reduction technique which tends to produce a sparse representation of data. Commonly, the error between the actual and recreated matrices is used as an objective function, but this method may not produce the type of representation we desire as it allows for the complexity of the model to grow, constrained only by the size of the subspace and the non-negativity requirement. If additional constraints, such as sparsity, are imposed the question of parameter selection becomes critical. Instead of adding sparsity constraints in an ad-hoc manner we propose a novel objective function created by using the principle of minimum description length (MDL). Our formulation, MDL-NMF, automatically trades off between the complexity and accuracy of the model using a principled approach with little parameter selection or the need for domain expertise. We demonstrate our model works effectively on three heterogeneous data-sets and on a range of semi-synthetic data showing the broad applicability of our method.