A Variational Autoencoder for Probabilistic Non-Negative Matrix Factorisation

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. Our approach uses a minimum description length methodology to provide a method for achieving automatic regularisation; as it is designed using neural networks it can leverage deep learning frameworks for automatic differentiation, fast gradient descent algorithms and GPU support. We demonstrate the effectiveness of PAE-NMF on three heterogeneous datasets: images, financial time series and genomic.