Query Complexity of k-NN based Mode Estimation

TL;DR

This paper introduces an adaptive querying algorithm for mode estimation in multivariate data using k-NN distances, achieving efficient query complexity with noisy oracle feedback.

Contribution

It proposes a novel sequential learning method that adaptively queries an oracle to identify the point with minimal k-NN distance, improving over existing approaches.

Findings

Achieves high-probability correctness with fewer queries

Provides instance-dependent upper bounds on query complexity

Demonstrates significant empirical improvements over baselines

Abstract

Motivated by the mode estimation problem of an unknown multivariate probability density function, we study the problem of identifying the point with the minimum k-th nearest neighbor distance for a given dataset of n points. We study the case where the pairwise distances are apriori unknown, but we have access to an oracle which we can query to get noisy information about the distance between any pair of points. For two natural oracle models, we design a sequential learning algorithm, based on the idea of confidence intervals, which adaptively decides which queries to send to the oracle and is able to correctly solve the problem with high probability. We derive instance-dependent upper bounds on the query complexity of our proposed scheme and also demonstrate significant improvement over the performance of other baselines via extensive numerical evaluations.