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url: https://openreview.net/pdf?id=LGgdb4TS4Z 

Local structural information can increase the adaptability of graph convolutional networks to large graphs with heterogeneous topology. Existing methods only use relatively simplistic topological information, such as node degrees.We present a novel approach leveraging advanced topological information, i.e., persistent homology, which measures the information flow efficiency at different parts of the graph. To fully exploit such structural information in real world graphs, we propose a new network architecture which learns to use persistent homology information to reweight messages passed between graph nodes during convolution. For node classification tasks, our network outperforms existing ones on a broad spectrum of graph benchmarks. 

Cardiac trabeculae are fine rod-like muscles whose ends are attached to the inner walls of ventricles. Accurate extraction of trabeculae is important yet challenging, due to the background noise and limited resolution of cardiac images. Existing works proposed to handle this task by modeling the trabeculae as topological handles for better extraction. Computing optimal representation of these handles is essential yet very expensive. In this work, we formulate the problem as a heuristic search problem, and propose novel heuristic functions based on advanced topological techniques. We show in experiments that the proposed heuristic functions improve the computation in both time and memory. 

Regularization plays a crucial role in supervised learning. Most existing methods enforce a global regularization in a structure agnostic manner. In this paper, we initiate a new direction and propose to enforce the structural simplicity of the classification boundary by regularizing over its topological complexity. In particular, our measurement of topological complexity incorporates the importance of topological features (e.g., connected components, handles, and so on) in a meaningful manner, and provides a direct control over spurious topological structures. We incorporate the new measurement as a topological penalty in training classifiers. We also propose an efficient algorithm to compute the gradient of such penalty. Our method provides a novel way to topologically simplify the global structure of the model, without having to sacrifice too much of the flexibility of the model. We demonstrate the effectiveness of our new topological regularizer on a range of synthetic and real-world datasets.