  Sparse dictionary learning (SDL) has become a popular method for adaptively
identifying parsimonious representations of a dataset, a fundamental problem in
machine learning and signal processing. While most work on SDL assumes a
training dataset of independent and identically distributed samples, a variant
known as convolutional sparse dictionary learning (CSDL) relaxes this
assumption, allowing more general sequential data sources, such as time series
or other dependent data. Although recent work has explored the statistical
properties of classical SDL, the statistical properties of CSDL remain
unstudied. This paper begins to study this by identifying the minimax
convergence rate of CSDL in terms of reconstruction risk, by both upper
bounding the risk of an established CSDL estimator and proving a matching
information-theoretic lower bound. Our results indicate that consistency in
reconstruction risk is possible precisely in the `ultra-sparse' setting, in
which the sparsity (i.e., the number of feature occurrences) is in $o(N)$ in
terms of the length N of the training sequence. Notably, our results make very
weak assumptions, allowing arbitrary dictionaries and dependent measurement
noise. Finally, we verify our theoretical results with numerical experiments on
synthetic data.
