Dynamic Maintenance of Kernel Density Estimation Data Structure: From Practice to Theory

TL;DR

This paper introduces a new theoretical framework for dynamically maintaining kernel density estimation data structures that are robust to adversarial queries, supporting efficient updates and adaptive querying.

Contribution

It presents the first theoretical framework for dynamic, robust KDE data structures with subquadratic space and sublinear update and query times.

Findings

Supports dynamic updates in sublinear time

Operates with subquadratic space complexity

Ensures robustness against adversarial queries

Abstract

Kernel density estimation (KDE) stands out as a challenging task in machine learning. The problem is defined in the following way: given a kernel function $f(x,y)$ and a set of points $\{x_1, x_2, \cdots, x_n \} \subset \mathbb{R}^d$, we would like to compute $\frac{1}{n}\sum_{i=1}^{n} f(x_i,y)$ for any query point $y \in \mathbb{R}^d$. Recently, there has been a growing trend of using data structures for efficient KDE. However, the proposed KDE data structures focus on static settings. The robustness of KDE data structures over dynamic changing data distributions is not addressed. In this work, we focus on the dynamic maintenance of KDE data structures with robustness to adversarial queries. Especially, we provide a theoretical framework of KDE data structures. In our framework, the KDE data structures only require subquadratic spaces. Moreover, our data structure supports the dynamic update of the dataset in sublinear time. Furthermore, we can perform adaptive queries with the potential adversary in sublinear time.