Learning Homotopy Inference in Euclidean Spaces and Riemannian Manifolds

Core Concepts

The authors extend the work of Niyogi, Smale, and Weinberger on homotopy learning to subsets of Euclidean spaces and Riemannian manifolds with positive reach. They provide tight bounds on the sampling parameters for successful homotopy inference.

Abstract

This article extends the seminal work by Niyogi, Smale, and Weinberger on homotopy learning to subsets of Euclidean spaces and Riemannian manifolds with positive reach. The authors establish conditions on sample quality parameters ε and δ that guarantee successful homotopy inference. By distinguishing between δ and ε, they provide explicit conditions for the union of balls to deformation-retract onto the underlying set or manifold. The results are shown to be optimal through explicit constructions in both Euclidean spaces and Riemannian manifolds with lower bounded sectional curvature.

Stats

We extend their results in terms of ambient space — we consider both Rd and Riemannian manifolds with lower bounded sectional curvature.
We distinguish two sample quality parameters — sample density ε and sample noisiness δ.
There exists a parameter r such that the union of balls deformation-retracts to S based on ε and δ.
For a specific choice of S and P, the homotopy of S is not inferrable from P if conditions on ε and δ are not satisfied.

Quotes

"We carefully distinguish the roles of δ and ε."
"This result expands on the work of Niyogi, Smale, and Weinberger."
"Our results can be applied to improve bounds also in this context."

Key Insights Distilled From

Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds

by Domi... at arxiv.org 03-01-2024

https://arxiv.org/pdf/2206.10485.pdf

Deeper Inquiries

How do these findings impact current practices in topological data analysis?

The findings presented in the research article have significant implications for current practices in topological data analysis (TDA). By extending and strengthening the work on learning homotopy types from samples of underlying spaces, the study provides a more comprehensive understanding of how to infer topology from partial geometric information. This advancement can enhance TDA techniques by offering tighter bounds and more general settings for analyzing datasets.
One key impact is the ability to reconstruct shapes or manifolds accurately from noisy or incomplete data. The tight bounds established in the study ensure that there exists a parameter r such that the union of balls centered at a sample deformation-retracts to the original shape. This capability can improve shape reconstruction algorithms and facilitate better inference of topological features from point cloud data.
Moreover, by distinguishing between sample density ε and sample noisiness δ, practitioners can adapt their analyses based on different levels of noise present in the dataset. This flexibility allows for more precise adjustments when dealing with real-world datasets that may contain varying degrees of noise.
Overall, these findings provide a solid foundation for advancing TDA methodologies, enabling researchers to extract meaningful insights about complex structures and shapes from limited observational data.

How could advancements in this field influence other areas beyond mathematics?

The advancements made in homotopy learning and topological data analysis have far-reaching implications beyond mathematics into various interdisciplinary fields:

Computer Vision: In computer vision applications like image segmentation, object recognition, or texture classification, understanding shapes and structures plays a crucial role. The methods developed for inferring topology from sampled data could enhance pattern recognition algorithms by providing robust shape reconstruction capabilities.

Medical Imaging: In medical imaging technologies such as MRI or CT scans, accurate reconstruction of anatomical structures is essential for diagnosis and treatment planning. The improved techniques for inferring topology could lead to better visualization tools and diagnostic accuracy in healthcare settings.

Robotics: Shape recognition is fundamental in robotics applications where robots interact with objects or navigate through environments autonomously. Advancements in topology inference could contribute to developing robots capable of perceiving their surroundings accurately based on sensor inputs.

Material Science: Understanding complex material structures at microscopic levels is vital for designing new materials with specific properties. Topological data analysis methods could aid researchers in characterizing material structures efficiently without requiring complete information about them.

By bridging mathematical concepts with practical applications across diverse fields, advancements in homotopy learning have the potential to revolutionize problem-solving approaches involving complex geometrical entities beyond traditional mathematical domains.