Sublinear Sketches for Approximate Nearest Neighbor and Kernel Density Estimation

TL;DR

This paper introduces new sublinear space and time sketching algorithms for approximate nearest neighbor search and kernel density estimation in dynamic data streams, with theoretical guarantees and practical validation.

Contribution

It presents the first sublinear sketches for ANN and A-KDE that balance memory, accuracy, and efficiency in streaming models, extending prior work on exact methods.

Findings

Achieves sublinear memory and query time for ANN in streaming models.

Provides the first sublinear sketch guarantee for A-KDE in the Sliding-Window model.

Experimental results show low error and lightweight sketches on real datasets.

Abstract

Approximate Nearest Neighbor (ANN) search and Approximate Kernel Density Estimation (A-KDE) are fundamental problems at the core of modern machine learning, with broad applications in data analysis, information systems, and large-scale decision making. In massive and dynamic data streams, a central challenge is to design compact sketches that preserve essential structural properties of the data while enabling efficient queries. In this work, we develop new sketching algorithms that achieve sublinear space and query time guarantees for both ANN and A-KDE for a dynamic stream of data. For ANN in the streaming model, under natural assumptions, we design a sublinear sketch that requires only $\mathcal{O}(n^{1+\rho-\eta})$ memory by storing only a sublinear ($n^{-\eta}$) fraction of the total inputs, where $\rho$ is a parameter of the LSH family, and $0<\eta<1$. Our method supports sublinear query time, batch queries, and extends to the more general Turnstile model. While earlier works have focused on Exact NN, this is the first result on ANN that achieves near-optimal trade-offs between memory size and approximation error. Next, for A-KDE in the Sliding-Window model, we propose a sketch of size $\mathcal{O}\left(RW \cdot \frac{1}{\sqrt{1+\epsilon} - 1} \log^2 N\right)$, where $R$ is the number of sketch rows, $W$ is the LSH range, $N$ is the window size, and $\epsilon$ is the approximation error. This, to the best of our knowledge, is the first theoretical sublinear sketch guarantee for A-KDE in the Sliding-Window model. We complement our theoretical results with experiments on various real-world datasets, which show that the proposed sketches are lightweight and achieve consistently low error in practice.