Convolutional Sparse Coding for Time Series Via a ℓ0 Penalty: an Efficient Algorithm with Statistical Guarantees

Identifying characteristic patterns in time series, such as heartbeats or brain responses to a stimulus, is critical to understanding the physical or physiological phenomena monitored with sensors. Convolutional sparse coding (CSC) methods, which aim to approximate signals by a sparse combination of short signal templates (also called atoms), are well-suited for this task. However, enforcing sparsity leads to non-convex and untractable optimization problems. This article proposes finding the optimal solution to the original and non-convex CSC problem when the atoms do not overlap. Specifically, we show that the reconstruction error satisfies a simple recursive relationship in this setting, which leads to an efficient detection algorithm. We prove that our method correctly estimates the number of patterns and their localization, up to a detection margin that depends on a certain measure of the signal-to-noise ratio. In a thorough empirical study, with simulated and real-world physiological data sets, our method is shown to be more accurate than existing algorithms at detecting the patterns' onsets.