  Convolutional dictionary learning (CDL), the problem of estimating
shift-invariant templates from data, is typically conducted in the absence of a
prior/structure on the templates. In data-scarce or low signal-to-noise ratio
(SNR) regimes, which have received little attention from the community, learned
templates overfit the data and lack smoothness, which can affect the predictive
performance of downstream tasks. To address this limitation, we propose GPCDL,
a convolutional dictionary learning framework that enforces priors on templates
using Gaussian Processes (GPs). With the focus on smoothness, we show
theoretically that imposing a GP prior is equivalent to Wiener filtering the
learned templates, thereby suppressing high-frequency components and promoting
smoothness. We show that the algorithm is a simple extension of the classical
iteratively reweighted least squares, which allows the flexibility to
experiment with different smoothness assumptions. Through simulation, we show
that GPCDL learns smooth dictionaries with better accuracy than the
unregularized alternative across a range of SNRs. Through an application to
neural spiking data from rats, we show that learning templates by GPCDL results
in a more accurate and visually-interpretable smooth dictionary, leading to
superior predictive performance compared to non-regularized CDL, as well as
parametric alternatives.
