Optimal bandwidth estimation for a fast manifold learning algorithm to detect circular structure in high-dimensional data
This work addresses the challenge of efficiently analyzing high-dimensional data for topological features, which is incremental as it builds on existing manifold learning techniques.
The paper tackles the problem of detecting circular topological structures in high-dimensional data by proposing a method to infer such structures from 2D projections using a fast manifold learning algorithm, while also estimating the optimal bandwidth parameter through minimization and providing limit theorems to characterize the behavior of these functions.
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 $σ$ 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.