Paper:
Learning brain regions via large-scale online structured sparse dictionary-learning
We propose a multivariate online dictionary-learning method for obtaining decompositions of brain images with structured and sparse components (aka atoms). Sparsity is to be understood in the usual sense: the dictionary atoms are constrained to contain mostly zeros. This is imposed via an 1 -norm constraint. By "structured", we mean that the atoms are piece-wise smooth and compact, thus making up blobs, as opposed to scattered patterns of activation. We propose to use a Sobolev (Laplacian) penalty to impose this type of structure. Combining the two penalties, we obtain decompositions that properly delineate brain structures from functional images. This non-trivially extends the online dictionary-learning work of Mairal et al. (2010), at the price of only a factor of 2 or 3 on the overall running time. Just like the Mairal et al. (2010) reference method, the online nature of our proposed algorithm allows it to scale to arbitrarily sized datasets. Preliminary xperiments on brain data show that our proposed method extracts structured and denoised dictionaries that are more intepretable and better capture inter-subject variability in small medium, and large-scale regimes alike, compared to state-of-the-art models.

Review:
This paper provides an online algorithm of learning structured dictionary to characterize neuroimaging data across individuals, by extending a well-established framework in [14] with a Sobolev prior to encourage spatial contiguity of dictionary atoms. The authors applied their method on empirical data, reported the effect of varying tuning parameters and compared the performance with two alternative methods in the field.   
Technical quality: I would rate marginally below 3 if given the option. The model definition was clear. The algorithms followed a well established framework [14], and appeared solid. However, maybe a bit undermined by its presentation, the paper did not seem to clearly demonstrate the empirical advantage of introducing the Sobolev prior in the results section. Especially, in Figure 2, it was not clear in what aspect the SSMF method was better than the two alternatives.  Was the spatial boundary more smooth quantitatively? Were there certain regions, which were known to be connected, only identified by SSMF? etc.  Besides (vaguely) saying results by SSMF were “more structured”, the authors could explicitly describe some criteria justified by neuroscience literature, and give quantitative measurements of the improvement according to these criteria.   Added after reading the rebuttal: The results provided in the rebuttal are very informative. I changed my rating on technical quality from 2 to 3. I hope the authors include these new results in their revision, and in addition, demonstrate that the atoms learned by SSMF are more structured (with quantitative measurements) and more meaningful (with reference to neuroscience literature).  Novelty/originality: There is notable original contribution in this paper, which is the extension of [14] that applied Sobolev priors to encourage spatially structured dictionary atoms.   Potential impact or usefulness:  Dictionary learning is a useful approach for explaining neuroimaging data across a large group of subjects, and introducing Sobolev prior or similar spatial priors should improve the results in theory. This paper provides an efficient online algorithm of doing so, and could add a good tool in the toolbox of the neuroscience/medical imaging community. The authors could also explore other applications of this method such as image segmentation and regional sensing.   Presentation: The general organization of the paper is okay. However, the paper appeared not to be a well-edited manuscript; there were confusing descriptions and typos that made it harder to understand the contents. I list below the confusing parts. Some of them seem to be easy-to-fix typos, but some may require more detailed explanations.   --In the matrix decomposition equation between Line 67 and 68,  should the transpose symbol on the group of $V^j$ (j = 1,..., r) be on each individual $V^j$’s? --Line 73, the reference of HCP dataset was missing.  --Line 83 “The Graph-Net penalty can imposes ...” --Line 146: bold sentence might be a co-author’s comment which was not fixed. --Last line of Equation (6) and Equation (8) did not agree on the sign $ V^j +A_t[j,j]^{-1}(VA_t^j -B_t^j)$  versus $A_t[j,j] V^j -(VA_t^j - B_t^j)$. Also in the last line of equation (6), I assume $V$ refers the dictionary in the last iteration, (i.e. the jth column is not the variable $v$ to optimize).  -- Line 157, “a single, a single” -- Line 169, “ We did an extensive study of th how” -- In Section 3.3, the paragraph near 147 just below Algorithm 2,  it would be nice if the authors explicitly gave the mathematical/algorithmic solution to Equation (6), even in the appendix.  -- Figure 2, the results by different methods were presented at different horizontal and sagittal slices, which made it hard to directly compare the results.  Based on the figure, I did not follow the description in Line 222 that claimed the results by SSMF model were more “structured”.  -- Figure 3, it was unclear what was predicted. Was it regressing brain data on behavioral metrics, if so, was it base on $U_i$’s? The legends were not explained in the main text nor the figure caption. The figure did not seem to show the PCA baseline mentioned in the caption.  I did not follow the description in Line 224-227, which claimed that R^2 greater than 0.1, (which curves should I look at?).