Topo-Geometric Analysis of Variability in Point Clouds Using Persistence Landscapes

Abstract

Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects known as persistence diagrams. To better aid statistical analysis, a functional representation of the diagrams, known as persistence landscapes, enable use of functional data analysis and machine learning tools. Topological and geometric variabilities inherent in point clouds are confounded in both persistence diagrams and landscapes, and it is important to distinguish topological signal from noise to draw reliable conclusions on the structure of the point clouds when using persistence homology. We develop a framework for decomposing variability in persistence diagrams into topological signal and topological noise through alignment of persistence landscapes using an elastic Riemannian metric. Aligned landscapes (amplitude) isolate the topological signal. Reparameterizations used for landscape alignment (phase) are linked to a resolution parameter used to generate persistence diagrams, and capture topological noise in the form of geometric, global scaling and sampling variabilities. We illustrate the importance of decoupling topological signal and topological noise in persistence diagrams (landscapes) using several simulated examples. We also demonstrate that our approach provides novel insights in two real data studies.