Topological Analysis of Kernel Density Estimates via Uncertainty-Aware Morse Complexes and Coresets

Authors

Weiran Lyu (University of Utah), Jeff M. Phillips (University of Utah), Bei Wang (University of Utah)

Presentation

Session

I'm not so certain

Time

Thursday, Nov 12, 08:00 – 08:12 (US/Eastern) · session 08:00 – 09:30

Location

Hall Essex north

Keywords

Uncertainty visualization, Morse complexes, density ridges, coresets, kernel density estimates, topological data analysis

Abstract

Large-scale spatial density datasets, such as galaxy distributions, crime incidents, disease outbreaks, and nighttime light intensity, often exhibit complex geometric and topological structure. A density ridge captures the backbone of high-density regions, where data concentrate and the density attains local maxima in directions orthogonal to the structure. Detecting and constructing such ridges is fundamental to applications in urban analytics, road networks, and cosmology. For example, density ridges of crime incidents reveal hotspots that inform patrol allocation, while ridges in galaxy distributions characterize the filamentary structure of the cosmic web. In this paper, we extract uncertainty-aware density ridges from kernel density estimates (KDEs) using Morse complexes. Rooted in Morse theory, Morse complexes provide a gradient-based topological abstraction of density fields. Their one-dimensional skeletons, referred to as Morse skeletons, connect saddle points to local maxima and naturally serve as representations of density ridges. However, these skeletons are sensitive to small perturbations in the data; certain regions exhibit pronounced instability, limiting their reliability as visualization and analysis primitives. To address this limitation, we quantify uncertainty in Morse skeletons and identify structurally stable regions that support the detection and characterization of filamentary structures. Our approach aggregates Morse complexes across an ensemble of perturbed datasets to produce an uncertainty heatmap. In addition, we introduce an optimized coreset construction for KDE that is compact while preserving the Morse complex structure of the original data. Through experiments on diverse applications, including patrol planning, traffic routing, and cosmic web reconstruction, we show that our uncertainty heatmaps characterize structural stability and enhance robustness, while our coresets enable efficient computation without compromising topological fidelity.

For Practitioners

This work is relevant to practitioners in urban analytics, transportation planning, computational cosmology, and scientific visualization who analyze large spatial point datasets. The proposed uncertainty-aware density ridge extraction method enables practitioners to assess the reliability of identified spatial structures, while the optimized coreset method accelerates kernel density estimation and topological analysis without compromising structural fidelity.