Dynamic Similarity Graph Construction with Kernel Density Estimation

TL;DR

This paper introduces a dynamic data structure for kernel density estimation and similarity graph maintenance, enabling efficient updates and spectral clustering in evolving datasets.

Contribution

It presents a novel dynamic data structure for KDE and similarity graphs, facilitating fast spectral clustering on data that changes over time.

Findings

Efficient dynamic maintenance of similarity graphs.

Fast spectral clustering on evolving datasets.

Validated on synthetic and real-world data.

Abstract

In the kernel density estimation (KDE) problem, we are given a set $X$ of data points in $\mathbb{R}^d$, a kernel function $k: \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$, and a query point $\mathbf{q} \in \mathbb{R}^d$, and the objective is to quickly output an estimate of $\sum_{\mathbf{x} \in X} k(\mathbf{q}, \mathbf{x})$. In this paper, we consider $\textsf{KDE}$ in the dynamic setting, and introduce a data structure that efficiently maintains the estimates for a set of query points as data points are added to $X$ over time. Based on this, we design a dynamic data structure that maintains a sparse approximation of the fully connected similarity graph on $X$, and develop a fast dynamic spectral clustering algorithm. We further evaluate the effectiveness of our algorithms on both synthetic and real-world datasets.