Fitting a Simplicial Complex using a Variation of k-means

TL;DR

This paper introduces a two-stage algorithm that generalizes k-means clustering to fit a simplicial complex to a point cloud, enabling effective approximation and dimension reduction.

Contribution

The paper presents a novel two-stage method combining a k-means-like fitting process with redundancy removal to approximate point clouds with simplicial complexes.

Findings

Effective approximation of point clouds by simplicial complexes

Dimension reduction achieved through the method

Simplifies complex data structures

Abstract

We give a simple and effective two stage algorithm for approximating a point cloud $\mathcal{S}\subset\mathbb{R}^m$ by a simplicial complex $K$. The first stage is an iterative fitting procedure that generalizes k-means clustering, while the second stage involves deleting redundant simplices. A form of dimension reduction of $\mathcal{S}$ is obtained as a consequence.