# Persistence Terrace for Topological Inference of Point Cloud Data

## TL;DR

The paper introduces the persistence terrace, a new topological summary plot for point cloud data that is robust, multi-scale, and parameter-free, enabling better inference of topological features across noise and scale.

## Contribution

It proposes the persistence terrace, a novel topological summary that overcomes limitations of existing methods by incorporating multiple smoothing parameters without losing scale information.

## Key findings

- Persistence terrace effectively isolates topological signals in noisy data.
- It allows inference of feature size and density across scales.
- Demonstrated on real muscle fiber data.

## Abstract

Topological data analysis (TDA) is a rapidly developing collection of methods for studying the shape of point cloud and other data types. One popular approach, designed to be robust to noise and outliers, is to first use a smoothing function to convert the point cloud into a manifold and then apply persistent homology to a Morse filtration. A significant challenge is that this smoothing process involves the choice of a parameter and persistent homology is highly sensitive to that choice; moreover, important scale information is lost. We propose a novel topological summary plot, called a persistence terrace, that incorporates a wide range of smoothing parameters and is robust, multi-scale, and parameter-free. This plot allows one to isolate distinct topological signals that may have merged for any fixed value of the smoothing parameter, and it also allows one to infer the size and point density of the topological features. We illustrate our method in some simple settings where noise is a serious issue for existing frameworks and then we apply it to a real data set by counting muscle fibers in a cross-sectional image.