  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.
