Learning the Morphology of Brain Signals Using Alpha-Stable Convolutional Sparse Coding
Neural time-series data contain a wide variety of prototypical signal waveforms (atoms) that are of significant importance in clinical and cognitive research. One of the goals for analyzing such data is hence to extract such 'shift-invariant' atoms. Even though some success has been reported with existing algorithms, they are limited in applicability due to their heuristic nature. Moreover, they are often vulnerable to artifacts and impulsive noise, which are typically present in raw neural recordings. In this study, we address these issues and propose a novel probabilistic convolutional sparse coding (CSC) model for learning shift-invariant atoms from raw neural signals containing potentially severe artifacts. In the core of our model, which we call αCSC, lies a family of heavy-tailed distributions called α-stable distributions. We develop a novel, computationally efficient Monte Carlo expectation-maximization algorithm for inference. The maximization step boils down to a weighted CSC problem, for which we develop a computationally efficient optimization algorithm. Our results show that the proposed algorithm achieves state-of-the-art convergence speeds. Besides, αCSC is significantly more robust to artifacts when compared to three competing algorithms: it can extract spike bursts, oscillations, and even reveal more subtle phenomena such as cross-frequency coupling when applied to noisy neural time series.
In this paper, the authors propose a novel probabilistic convolutional sparse coding model (alpha-stable CSC) for learning shift-invariant atoms from raw neural signals. They propose an inference strategy based on Monte Carlo EM to deal efficiently with heavy tailed noise and take into account the polarity of neural activations with a positivity constraint. The formulation allows the use of fast quasi-Newton methods for the M-step which outperform previously proposed state-of-the-art ADMM based algorithms. The experiment results on LFP data demonstrate that such algorithms can be robust to the presence of transient artifacts in data and reveal insights on neural time-series without supervision.
The main contribution of this study is extending a general (alpha-stable distribution) framework, comparing to the Gaussian distribution. From experiment point of view, the proposed approach is very useful. However, there are a few comments and suggestions for the authors to consider:
(1)	Figure 1(a) only demonstrates the PDFs of alpha-stable distributions characterized by alpha and beta. Please try to describe the characters of the scale parameter and location parameter. 
(2)	As written in the paper, “As an important special case of the alpha-stable distributions, we obtain the Gaussian distribution when alpha= 2 and beta = 0”, the authors emphasize that the current CSC models might be limited as they consider an L2 reconstruction error. While the extending model is more robust to corrupted data than the Gaussian models. However, I wonder if the experiments should include Gaussian Mixture Model as a fair comparison.
(3)	How to determine the parameters (i.e., lambda, N, T, L, K), respectively, in synthetic and LEP data experiments?