We propose PLLay, a novel topological layer for general deep learning models based on persistence landscapes, in which we can efficiently exploit the underlying topological features of the input data structure. In this work, 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 PLLay is learned during training via backpropagation, without requiring any input featurization or data preprocessing. We provide a novel adaptation for the DTM function-based filtration, and show that the proposed layer is robust against noise and outliers through a stability analysis. We demonstrate the effectiveness of our approach by classification experiments on various datasets.
Topological descriptors of foot clearance gait dynamics improve differential diagnosis of Parkinsonism
Centre of Mathematics (CMAT) School of Sciences University of Minho（数学中心（CMAT）科学学院矿产大学）；Neurology Departmet Hospital da Senhora da Oliveira（神经科部门圣奥古斯丁医院）；ICVS School of Medicine University of Minho（ICVS医学院矿产大学）；Algoritmi Centre School of Engineering University of Minho（算法中心工程学院矿产大学）；Centre of Mathematics (CMUP) University of Porto（数学中心（CMUP）波尔图大学）；School and Economics and Management University of Porto（学校和经济管理波尔图大学）
A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science
a Department of Mathematics, Colorado State University, Fort Collins, CO b Dept.\ of Electrical；Computer Engineering, Colorado State University, Fort Collins, CO c Cooperative Institute for Research in the Atmosphere (CIRA), Colorado State University, Fort Collins, CO
