Multivariate Convolutional Sparse Coding for Electromagnetic Brain Signals
Frequency-specific patterns of neural activity are traditionally interpreted as sustained rhythmic oscillations, and related to cognitive mechanisms such as attention, high level visual processing or motor control. While alpha waves (8-12 Hz) are known to closely resemble short sinusoids, and thus are revealed by Fourier analysis or wavelet transforms, there is an evolving debate that electromagnetic neural signals are composed of more complex waveforms that cannot be analyzed by linear filters and traditional signal representations. In this paper, we propose to learn dedicated representations of such recordings using a multivariate convolutional sparse coding (CSC) algorithm. Applied to electroencephalography (EEG) or magnetoencephalography (MEG) data, this method is able to learn not only prototypical temporal waveforms, but also associated spatial patterns so their origin can be localized in the brain. Our algorithm is based on alternated minimization and a greedy coordinate descent solver that leads to state-of-the-art running time on long time series. To demonstrate the implications of this method, we apply it to MEG data and show that it is able to recover biological artifacts. More remarkably, our approach also reveals the presence of non-sinusoidal mu-shaped patterns, along with their topographic maps related to the somatosensory cortex.
This work extends convolutional sparse coding to the multivariate case with a focus on multichannel EEG decomposition. This corresponds to a non-convex minimization problem and a local minimum is found via an alternating optimization. Reasonable efficient bookkeeping (precomputation of certain factors, and windowing for locally greedy coordinate descent) is used to improve scalability. The locally greedy coordinate descent cycles time windows, but computes a greedy coordinate descent within each window.  As spatial patterns are essential for understanding EEG, this multivariate extension is an important contribution. The work is clearly presented. The results demonstrate an efficient implementation. Some preliminary experiments show the potential for automatically learning brain waveforms.   Weaknesses:  In previous work, Jas et al. [13] consider different noise whereas this model assumes Gaussian noise versus the alpha stable noise. The authors should comment how this work could be extended to a different noise model.  Parameter/model selection ($L$ and $K$) are not discussed for the real-world signal (lines 235–253). Choosing $K$ too large can lead to cases where some atoms are never used or updated, and other cases where the same waveform appears as different atoms (perhaps at different shifts). Perspectives on these real-world considerations should be mentioned. All the learned waveforms should be reported in the supplement.   There are obvious similarities between greedy coordinate descent and the well-known matching pursuit algorithm, which is not mentioned. With the non-negative restriction, matching pursuit for time-series would have the same update in Eq. 5 without the shrinkage by lambda on the numerator. Assuming unit-norm patterns the greedy coordinate descent strategy (Eq. 7) would match the matching pursuit selection of maximal inner-product. Thus, the only change is that matching pursuit is done without shrinkage (lambda). Using matching pursuit, the remainder of the multivariate framework would not need to change. However, to be as efficient as the locally greedy version, a windowed version of matching pursuit should be done, which itself would be a novel algorithm. At least the issues with $\lambda_{max}$ could be replaced by using a fixed cardinality on the support. The authors could consider a discussion of the relation to matching pursuit as it has been used for both template matching (including the multivariate case) and learning waveforms in neural signals. Also to facilitate future comparisons the cardinality of support of the sparse code for different values of lambda could be investigated and reported.  Relevant reference: Piotr J. Durka, Artur Matysiak, Eduardo Martínez Montes, Pedro Valdés Sosa, Katarzyna J. Blinowska, Multichannel matching pursuit and EEG inverse solutions, Journal of Neuroscience Methods, Volume 148, Issue 1, 2005, Pages 49-59.  Based on my understanding of the reasoning, the authors believe the improved convergence efficiencies of a complete greedy coordinate descent would not compensate for the increased computational complexity of the complete greedy search. However, the gap between the local and globally greedy version could be narrowed: Couldn't the algorithm be improved by recomputing the differences in Eq. 7 after the update in each window, keeping them in a priority queue, and then proceeding to choose the best window? This could be implemented after the first loop through the $M$ windows in Algorithm 1.  Lines 218–227 How are the spatial maps in this example created? From the text it appears they are distinct for the two atoms. But the multivariate approach still would be better than single channel even if the spatial maps were the same due to spatial independence of noise.  This could be clarified.  Figure 4. How was this atom selected (by manual inspection)? Is it representative or simply the most mu-like waveform? How often does this atom appear? Are there similar atoms to it with different spatial patterns?    Minor issues: Line 20 "is central" is perhaps an overstatement in the context of neuroscience. "can be used to investigate" would be more reasonable. Line 22 duration -> durations Line 83 "aka" Line 98 overlapping -> overlapped Line 104 multivariate with -> multivariate CSC with Equations 2, 3 & 4 Should indicate the full sets of parameters as arguments in the minimizations. Especially to distinguish the case for a single multivariate signal in Equation 4.  Line 246 "9 Hz.," Line 252 "In notably includes" Line 178 values -> value Line 179 "allows to use" Line 335 'et al.' in author list for reference [29] without additional authors. Line 356 'et al.' in author list for reference [38] with only one more author.  ----- The authors have answered many of my questions. I look forward to a final version that will incorporate the additional discussion.