Paper:
Structure-Aware Convolutional Neural Networks
Convolutional neural networks (CNNs) are inherently subject to invariable filters that can only aggregate local inputs with the same topological structures. It causes that CNNs are allowed to manage data with Euclidean or grid-like structures (e.g., images), not ones with non-Euclidean or graph structures (e.g., traffic networks). To broaden the reach of CNNs, we develop structure-aware convolution to eliminate the invariance, yielding a unified mechanism of dealing with both Euclidean and non-Euclidean structured data. Technically, filters in the structure-aware convolution are generalized to univariate functions, which are capable of aggregating local inputs with diverse topological structures. Since infinite parameters are required to determine a univariate function, we parameterize these filters with numbered learnable parameters in the context of the function approximation theory. By replacing the classical convolution in CNNs with the structure-aware convolution, Structure-Aware Convolutional Neural Networks (SACNNs) are readily established. Extensive experiments on eleven datasets strongly evidence that SACNNs outperform current models on various machine learning tasks, including image classification and clustering, text categorization, skeleton-based action recognition, molecular activity detection, and taxi flow prediction.

Review:
The paper proposes a structure-aware convolution structure that can naturally aggregate local inputs from diverse topological structures without predefining data structures. The learnable filters in the convolution operation are well designed in a simple way. The methods shows better quantitative results on eleven datasets when comparing to baselines.  Strength: 1. Simple. The convolution operator is simply unified on diverse topological structures. 2. Extensive experiments on six Euclidean and five non-Euclidean structured datasets.  3. Results show the proposed method has certain improvements over recent convolution neural network and graph convolution networks.  4. The paper is well written, and clear to my understanding.   Weakness: 1. Notation confusion. In equation (1) and theorem #1, x_i denotes a local input from (i-m)-th vertex to (i+m)-th vertex. However, in equation (6),  a same x_i represents a single value (with multi-channel) at i-th vertex only. 2. The detailed information for graph construction is missing, e.g. the way to define edges and construct a graph for non-euclidean datasets. 3. Despite the proposed method shares a different idea with [17], the P_i matrix in theorem #1 mathematically transforms the input x_i into a high-dimensional space where a convolution weights v are applied. Can the authors briefly compares the proposed method with [17]? My feeling is that [17] should be a good baseline to compare on Euclidean datasets.  4. It seems that the baselines in Experiment 5.2 are not the original models, but the authors' own implementations, e.g. ActCNNs[18] has 93% accuracy on Cifar10 instead of 90%. Can the authors comment about this?  5. It would be good if the author can compare number of parameters between the proposed network and used baselines.   Minor comments: 1. In figure 3(f), the improvement looks saturated when using 56 layers network. Will the proposed method also benefit deeper network (e.g. resnet110 in Cifar10)?  2. It would be interesting if the authors would comment on how their approach can be applied to point cloud dataset.  Conclusion: I hope the authors will response the above questions in the rebuttal and include these in the final version. Given the novelty of the approach and extensive experiments, the submission would be stronger if the authors could fix these flaws.