Efficient Topological Layer based on Persistent Landscapes

We propose LandLayer, a novel topological layer for general deep learning models based on persistence landscapes, in which we can efficiently exploit underlying topological features of the input data structure. We show differentiability with respect to layer inputs, for a general persistent homology with arbitrary filtration. Thus, our proposed layer can be placed anywhere in the network and feed critical information on the topological features of input data into subsequent layers to improve the learnability of the networks toward a given task. A task-optimal structure of LandLayer is learned during training via backpropagation, without requiring any input featurization or data preprocessing. We provide novel stability results, including an adaptation for the robust DTM filtration function, and show that the proposed layer is robust against noise and outliers. We demonstrate the effectiveness of our approach by classification experiments on various datasets.