Towards Persistence-Based Reconstruction in Euclidean Spaces

TL;DR

This paper introduces a new persistence-based reconstruction method for Euclidean spaces that efficiently recovers the homology of smooth submanifolds with complexity depending on the intrinsic dimension, improving practicality in higher dimensions.

Contribution

The authors propose a novel approach that scales with the intrinsic dimension, providing a more practical method for manifold reconstruction in higher-dimensional Euclidean spaces.

Findings

Method retrieves homology in time at most c(m)n^5 for dense samples from smooth m-submanifolds.

New theoretical results on Čech, Rips, and witness complex filtrations in Euclidean spaces.

Applicable to a wide range of compact subsets with varying complexities.

Abstract

Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes of compact subsets of Euclidean space $\R^d$. However, the complexities of these methods scale up exponentially with d, which makes them impractical in medium or high dimensions, even for handling low-dimensional submanifolds. In this paper, we introduce a novel approach that stands in-between classical reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data. Specifically, when the data points are sufficiently densely sampled from a smooth $m$-submanifold of $\R^d$, our method retrieves the homology of the submanifold in time at most $c(m)n^5$, where $n$ is the size of the input and $c(m)$ is a constant depending solely on $m$. It can also provably well handle a wide range of compact subsets of $\R^d$, though with worse complexities. Along the way to proving the correctness of our algorithm, we obtain new results on \v{C}ech, Rips, and witness complex filtrations in Euclidean spaces.