Optimal bandwidth estimation for a fast manifold learning algorithm to detect circular structure in high-dimensional data

TL;DR

This paper introduces a method to detect circular structures in high-dimensional data using a fast manifold learning approach, and also provides a way to estimate the optimal bandwidth parameter for improved detection accuracy.

Contribution

It offers a novel framework for inferring topological circularity and estimating optimal bandwidth in high-dimensional data analysis via manifold learning.

Findings

Method effectively detects circular structures in high-dimensional data.

Provides theoretical limit theorems for bandwidth behavior.

Estimates optimal bandwidth for manifold learning applications.

Abstract

We provide a way to infer about existence of topological circularity in high-dimensional data sets in $\mathbb{R}^d$ from its projection in $\mathbb{R}^2$ obtained through a fast manifold learning map as a function of the high-dimensional dataset $\mathbb{X}$ and a particular choice of a positive real $\sigma$ known as bandwidth parameter. At the same time we also provide a way to estimate the optimal bandwidth for fast manifold learning in this setting through minimization of these functions of bandwidth. We also provide limit theorems to characterize the behavior of our proposed functions of bandwidth.