Unsupervised Learning of Density Estimates with Topological Optimization

TL;DR

This paper introduces a topology-based loss function for unsupervised kernel density estimation, automating bandwidth selection and improving the preservation of topological features across various dimensions.

Contribution

It presents a novel unsupervised method leveraging topological data analysis to optimize kernel bandwidth without manual tuning.

Findings

Outperforms classical bandwidth selection methods

Effectively captures topological features in high-dimensional data

Demonstrates robustness across different data dimensions

Abstract

Kernel density estimation is a key component of a wide variety of algorithms in machine learning, Bayesian inference, stochastic dynamics and signal processing. However, the unsupervised density estimation technique requires tuning a crucial hyperparameter: the kernel bandwidth. The choice of bandwidth is critical as it controls the bias-variance trade-off by over- or under-smoothing the topological features. Topological data analysis provides methods to mathematically quantify topological characteristics, such as connected components, loops, voids et cetera, even in high dimensions where visualization of density estimates is impossible. In this paper, we propose an unsupervised learning approach using a topology-based loss function for the automated and unsupervised selection of the optimal bandwidth and benchmark it against classical techniques -- demonstrating its potential across different dimensions.