Geometry-driven collapses for converting a Cech complex into a triangulation of a nicely triangulable shape

TL;DR

This paper proves that under certain conditions, the Rips complex of a sampled shape can be simplified through collapses into a triangulation of the shape, providing a theoretical basis for a practical simplification method.

Contribution

It offers a theoretical justification for converting Rips complexes into shape triangulations via collapses, assuming the shape is nicely triangulable and well-sampled.

Findings

Rips complex can be transformed into a shape homeomorphic complex via collapses.

The process relies on converting Rips to Cech complexes and then to a nerve of a robust covering.

Theoretical results support empirical observations of collapse-based simplification.

Abstract

Given a set of points that sample a shape, the Rips complex of the data points is often used in machine-learning to provide an approximation of the shape easily-computed. It has been proved recently that the Rips complex captures the homotopy type of the shape assuming the vertices of the complex meet some mild sampling conditions. Unfortunately, the Rips complex is generally high-dimensional. To remedy this problem, it is tempting to simplify it through a sequence of collapses. Ideally, we would like to end up with a triangulation of the shape. Experiments suggest that, as we simplify the complex by iteratively collapsing faces, it should indeed be possible to avoid entering a dead end such as the famous Bing's house with two rooms. This paper provides a theoretical justification for this empirical observation. We demonstrate that the Rips complex of a point-cloud (for a well-chosen scale parameter) can always be turned into a simplicial complex homeomorphic to the shape by a sequence of collapses, assuming the shape is nicely triangulable and well-sampled (two concepts we will explain in the paper). To establish our result, we rely on a recent work which gives conditions under which the Rips complex can be converted into a Cech complex by a sequence of collapses. We proceed in two phases. Starting from the Cech complex, we first produce a sequence of collapses that arrives to the Cech complex, restricted by the shape. We then apply a sequence of collapses that transforms the result into the nerve of some robust covering of the shape.