Topological Data Analysis for Object Data

Abstract

Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using persistence landscapes to vectorize object data and perform statistical analysis. We apply to this pipeline to some biological images that were previously shown to be challenging to study using shape theory. Surprisingly, the most persistent features are shown to be "topological noise" and the statistical analysis depends on the less persistent features which we refer to as the "geometric signal". We also describe the first steps to a new approach to using topology for object data analysis, which applies topology to distributions on object spaces.

arXiv comment16 pages, 12 figures

References in corpus (6)

• Statistical topological data analysis using persistence landscapes
• Confidence sets for persistence diagrams
• A persistence landscapes toolbox for topological statistics
• Directions and projective shapes
• Using persistent homology and dynamical distances to analyze protein binding
• Testing for the Equality of two Distributions on High Dimensional Object Spaces

Cited by in corpus (6)

• The persistence landscape and some of its properties
• Persistent homology detects curvature
• Topological Learning for Brain Networks
• A persistent homology approach to heart rate variability analysis with an application to sleep-wake classification
• The Weighted Euler Curve Transform for Shape and Image Analysis
• Anti-MANOVA on Compact Manifolds with Applications to 3D Projective Shape Analysis