On the Reconstruction Risk of Convolutional Sparse Dictionary Learning

Sparse dictionary learning (SDL) has become a popular method for learning parsimonious representations of data, a fundamental problem in machine learning and signal processing. While most work on SDL assumes a training dataset of independent and identically distributed (IID) samples, a variant known as convolutional sparse dictionary learning (CSDL) relaxes this assumption to allowdependent, non-stationary sequential data sources. Recent work has explored statistical properties of IID SDL; however, the statistical properties of CSDL remain largely unstudied. This paper identifies minimax rates of CSDL in terms of reconstruction risk, providing both lower and upper bounds in a variety of settings. Our results make minimal assumptions, allowing arbitrary dictionaries and showing that CSDL is robust to dependent noise. We compare our results to similar results for IID SDL and verify our theory with synthetic experiments.