  Given a continuous-time signal that can be modeled as the superposition of
localized, time-shifted events from multiple sources, the goal of Convolutional
Dictionary Learning (CDL) is to identify the location of the events--by
Convolutional Sparse Coding (CSC)--and learn the template for each source--by
Convolutional Dictionary Update (CDU). In practice, because we observe samples
of the continuous-time signal on a uniformly-sampled grid in discrete time,
classical CSC methods can only produce estimates of the times when the events
occur on this grid, which degrades the performance of the CDU. We introduce a
CDL framework that significantly reduces the errors arising from performing the
estimation in discrete time. Specifically, we construct an expanded dictionary
that comprises, not only discrete-time shifts of the templates, but also
interpolated variants, obtained by bandlimited interpolation, that account for
continuous-time shifts. For CSC, we develop a novel computationally efficient
CSC algorithm, termed Convolutional Orthogonal Matching Pursuit with
interpolated dictionary (COMP-INTERP). We benchmarked COMP-INTERP to
Contiunuous Basis Pursuit (CBP), the state-of-the-art CSC algorithm for
estimating off-the-grid events, and demonstrate, on simulated data, that 1)
COMP-INTERP achieves a similar level of accuracy, and 2) is two orders of
magnitude faster. For CDU, we derive a novel procedure to update the templates
given sparse codes that can occur both on and off the discrete-time grid. We
also show that 3) dictionary update with the overcomplete dictionary yields
more accurate templates. Finally, we apply the algorithms to the spike sorting
problem on electrophysiology recording and show their competitive performance.
