# Unsupervised Learning of Density Estimates with Topological Optimization

arXiv: https://arxiv.org/abs/2512.08895 · HTML: https://mlsanity.com/paper/2512.08895/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 37 references that are in the map.
- First posted on arXiv 9 Dec 2025.

## Builds on (most cited first, 4 of 4)

- [Robust Topological Inference: Distance To a Measure and Kernel Distance](https://mlsanity.com/paper/1412.7197/) (2014)
- [Modal Regression using Kernel Density Estimation: a Review](https://mlsanity.com/paper/1710.07004/) (2017)
- [Topological density estimation](https://mlsanity.com/paper/1701.09025/) (2017)
- [A Divide-and-Conquer Approach to Persistent Homology](https://mlsanity.com/paper/2410.01839/) (2024)

## Abstract (arXiv)

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.