Gaussian processes are the leading class of distributions on random
functions, but they suffer from well known issues including difficulty scaling
and inflexibility with respect to certain shape constraints (such as
nonnegativity). Here we propose Deep Random Splines, a flexible class of random
functions obtained by transforming Gaussian noise through a deep neural network
whose output are the parameters of a spline. Unlike Gaussian processes, Deep
Random Splines allow us to readily enforce shape constraints while inheriting
the richness and tractability of deep generative models. We also present an
observational model for point process data which uses Deep Random Splines to
model the intensity function of each point process and apply it to neuroscience
data to obtain a low-dimensional representation of spiking activity. Inference
is performed via a variational autoencoder that uses a novel recurrent encoder
architecture that can handle multiple point processes as input.

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