New Bounds for Kernel Sums via Fast Spherical Embeddings

2026-05-02

AI Summary

This work addresses the inefficiency of existing algorithms for kernel mean estimation in high-dimensional settings when the error tolerance ε is small and the data diameter Δ is moderate. To overcome this limitation, the authors propose a novel fast spherical embedding technique that preserves local Euclidean distances while effectively controlling the global diameter of the embedded data, thereby mitigating distance collapse in high dimensions. This approach achieves, for the first time, a joint optimization of local distance preservation and global diameter constraints. As a result, it establishes a new upper bound on the time complexity for kernel mean estimation of Õ(d + εΔ² + 1/ε³), which significantly improves upon prior bounds of O(d/ε²), Õ(d + 1/ε⁴), and Õ(d + Δ²/ε²) in regimes with small ε and moderate Δ.

Abstract

We study query time bounds for the fundamental problem of estimating the kernel mean $\frac1{|X|}\sum_{x\in X}\mathbf{k}(x,y)$ of a query $y$ in a finite dataset $X\subset\mathbb{R}^d$ up to a prescribed additive error $\varepsilon$. The best known bounds for the Gaussian kernel are $O(d/\varepsilon^2)$, $\widetilde O(d+1/\varepsilon^4)$, and $\widetilde O(d+Δ^2/\varepsilon^2)$, where $Δ$ is the diameter of a region containing the points. We prove the new bound $\tilde O(d+\varepsilonΔ^2+1/\varepsilon^3)$, which improves over the previous ones in regimes with small error $\varepsilon$ and intermediate diameter $Δ$. At the center of our proof is a new fast spherical embedding theorem in the sense introduced by Bartal, Recht and Schulman (2011), which limits the embedded data diameter while preserving local Euclidean distances and avoiding ``distance collapse'' at larger scales. This fast embedding theorem may be of independent interest.

Problem

• kernel mean estimation
• query time bounds
• Gaussian kernel
• additive error
• spherical embeddings

Innovation

• fast spherical embedding
• kernel mean estimation
• Gaussian kernel
• query time bounds
• distance preservation