Beyond kNN: Adaptive, Sparse Neighborhood Graphs via Optimal Transport

TL;DR

This paper introduces an adaptive method for constructing neighborhood graphs using optimal transport, which adjusts to data density variations and improves performance in learning tasks.

Contribution

It presents a novel, parameter-efficient approach to build adaptive neighborhood graphs via quadratically regularised optimal transport, addressing limitations of fixed kNN graphs.

Findings

Graphs outperform traditional kNN in various tasks

Method adapts to data density and noise levels

Improves unsupervised and semi-supervised learning results

Abstract

Nearest neighbour graphs are widely used to capture the geometry or topology of a dataset. One of the most common strategies to construct such a graph is based on selecting a fixed number k of nearest neighbours (kNN) for each point. However, the kNN heuristic may become inappropriate when sampling density or noise level varies across datasets. Strategies that try to get around this typically introduce additional parameters that need to be tuned. We propose a simple approach to construct an adaptive neighbourhood graph from a single parameter, based on quadratically regularised optimal transport. Our numerical experiments show that graphs constructed in this manner perform favourably in unsupervised and semi-supervised learning applications.