Reconstruction of Manifold Distances from Noisy Observations

2025-11-17

AI Summary

This paper addresses the problem of recovering the intrinsic geometry—specifically, the true geodesic distances—of a one-dimensional unit-diameter manifold from noisy pairwise distance observations under dense sampling. We propose a robust distance recovery framework that abandons classical assumptions of i.i.d. additive noise with known moments, instead accommodating broader noise models and missing observations. Our method employs a geometrically aware clustering scheme built upon nonparametric estimation and the L² norm, integrating constraints from manifold curvature and injectivity radius for principled geometric inference. Theoretically, we establish that when the sampling density satisfies Ω(ε⁻²ᵈ⁻² log(1/ε)), the estimated distances achieve an additive error bound of O(ε log ε⁻¹), with computational complexity sub-cubic in the number of points (o(N³)). This significantly improves upon state-of-the-art approaches in both statistical accuracy and scalability.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Distributed Search
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We consider the problem of reconstructing the intrinsic geometry of a manifold from noisy pairwise distance observations. Specifically, let $M$ denote a diameter 1 d-dimensional manifold and $μ$ a probability measure on $M$ that is mutually absolutely continuous with the volume measure. Suppose $X_1,dots,X_N$ are i.i.d. samples of $μ$ and we observe noisy-distance random variables $d'(X_j, X_k)$ that are related to the true geodesic distances $d(X_j,X_k)$. With mild assumptions on the distributions and independence of the noisy distances, we develop a new framework for recovering all distances between points in a sufficiently dense subsample of $M$. Our framework improves on previous work which assumed i.i.d. additive noise with known moments. Our method is based on a new way to estimate $L_2$-norms of certain expectation-functions $f_x(y)=mathbb{E}d'(x,y)$ and use them to build robust clusters centered at points of our sample. Using a new geometric argument, we establish that, under mild geometric assumptions--bounded curvature and positive injectivity radius--these clusters allow one to recover the true distances between points in the sample up to an additive error of $O(varepsilon log varepsilon^{-1})$. We develop two distinct algorithms for producing these clusters. The first achieves a sample complexity $N asymp varepsilon^{-2d-2}log(1/varepsilon)$ and runtime $o(N^3)$. The second introduces novel geometric ideas that warrant further investigation. In the presence of missing observations, we show that a quantitative lower bound on sampling probabilities suffices to modify the cluster construction in the first algorithm and extend all recovery guarantees. Our main technical result also elucidates which properties of a manifold are necessary for the distance recovery, which suggests further extension of our techniques to a broader class of metric probability spaces.

Problem

• Reconstructing intrinsic manifold geometry from noisy distance observations
• Developing robust clustering methods to recover true geodesic distances
• Handling missing observations and extending to metric probability spaces

Innovation

• Estimating L2-norms of expectation-functions for robust clustering
• Using geometric arguments to recover distances with bounded error
• Developing two algorithms with improved sample complexity and runtime