Denoising Distances in Metric Measure Spaces

2026-06-16

AI Summary

This study addresses the problem of recovering true pairwise distances from noisy observations and identifying local clusters in general metric measure spaces under low regularity assumptions. To this end, the authors propose a near-linear-time algorithm that efficiently extracts large-scale local clusters around each sampled point at a fixed accuracy level for distance denoising, alongside a more computationally expensive algorithm achieving higher precision. This work is the first to extend distance denoising and clustering to general metric measure spaces, revealing a statistical–computational trade-off absent in the Riemannian manifold setting: while fixed-accuracy recovery can be achieved efficiently, attaining higher accuracy necessarily incurs greater computational cost.

Abstract

Recent work studied the problem of finding clusters and denoising pairwise distances from noisy distances of points sampled on a manifold. We study the same problems in more general metric measure spaces under \lowerphiregularity{}. We give an algorithm that extracts large localized clusters around every sampled point and uses them to denoise distances to any fixed accuracy, with near-linear running time in the dense fixed-accuracy regime. We also show how to achieve much higher accuracy with a non-efficient algorithm. This suggests that unlike the Riemannian case, denoising to higher accuracy in more general metric spaces has a statistical-computational gap.

Problem

• denoising
• metric measure spaces
• clustering
• noisy distances
• statistical-computational gap

Innovation

• metric measure spaces
• distance denoising
• localized clustering
• statistical-computational gap
• phi-regularity