Denoising distances beyond the volumetric barrier

Abstract: We study the problem of reconstructing the latent geometry of a $d$-dimensional Riemannian manifold from a random geometric graph. While recent works have made significant progress in manifold recovery from random geometric graphs, and more generally from noisy distances, the precision of pairwise distance estimation has been fundamentally constrained by the volumetric barrier, namely the natural sample-spacing scale $n^{-1/d}$ coming from the fact that a generic point of the manifold typically lies at distance of order $n^{-1/d}$ from the nearest sampled point. In this paper, we introduce a novel approach, Orthogonal Ring Distance Estimation Routine (ORDER), which achieves a pointwise distance estimation precision of order $n^{-2/(d+5)}$ up to polylogarithmic factors in $n$ in polynomial time. This strictly beats the volumetric barrier for dimensions $d > 5$. As a consequence of obtaining pointwise precision better than $n^{-1/d}$, we prove that the Gromov--Wasserstein distance between the reconstructed metric measure space and the true latent manifold is of order $n^{-1/d}$. This matches the Wasserstein convergence rate of empirical measures, demonstrating that our reconstructed graph metric is asymptotically as good as having access to the full pairwise distance matrix of the sampled points. Our results are proven in a very general setting which includes general models of noisy pairwise distances, sparse random geometric graphs, and unknown connection probability functions.

• The paper presents ORDER, a polynomial-time algorithm that achieves an error bound of O(n^(-2/(d+5))) for dimensions d > 5, breaking the volumetric barrier.
• It leverages a unique orthogonal ring-based clustering method to reduce bias from local sample spacing, resulting in quadratic error reduction.
• The approach is robust to noise and sparsity, reconstructing metric measure spaces optimally in the Gromov–Wasserstein sense.

Denoising Pairwise Distances in Random Geometric Graphs Beyond the Volumetric Barrier

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Problem Setting and Motivation

This paper addresses the fundamental problem of reconstructing latent pairwise distances among $n$ points randomly sampled from a $d$-dimensional Riemannian manifold $M$, based only on a random geometric graph whose edge connections are determined via a monotone link function $p$ of the underlying metric. The key difficulty inherent to this recovery problem is the so-called volumetric barrier: due to the geometry of random samples, the minimal possible error for estimating pairwise distances via local neighborhood information is of order $n^{-1/d}$—the typical nearest-neighbor distance. All previous recovery schemes—including those operating with full or noisy distance matrices or in the presence of unknown link functions—are stymied by this statistical threshold.

Main Contributions

The paper introduces a new polynomial-time algorithm, ORDER (Orthogonal Ring Distance Estimation Routine), which—remarkably—breaks through the volumetric barrier for all $d > 5$. Specifically, ORDER achieves an error bound of $O( n^{-2/(d+5)} )$ up to polylogarithmic factors in $n$. This is strictly better than $n^{-1/d}$ at high dimension. The algorithm operates only under minimal regularity assumptions on $M$, the sampling measure $d$0, and the link function $d$1, and accommodates both sparse and weighted graphs, as well as unknown $d$2.

Crucially, the authors prove that, leveraging error smaller than the volumetric scale, the reconstructed metric measure space (in the Gromov–Wasserstein sense) is lost only up to the $d$3 benchmarking rate, matching optimal empirical measure convergence.

Summary of claims:

• For $d$4, there exists a polynomial-time estimator of pairwise distances with error $d$5 (polylogarithmic factors suppressed), strictly outperforming the volumetric barrier.
• The method remains valid in the setting of noisy distances, unknown link functions (recovering distances up to an unknown overall scale), and sparse geometric graphs with $d$6.
• The induced metric measure space matches the Wasserstein rate of the empirical distribution: $d$7.

Technical Contents

Breaking the Volumetric Barrier

Traditional manifold learning and graph recovery methods rely on local clustering and averaging, where clusters are formed at a radius $d$8 and distances are estimated either by neighborhood overlap or bipartite subgraph statistics, yielding the effective limit of $d$9 for the distance error—never outperforming $M$0.

ORDER circumvents this limitation by adapting a pair-dependent cluster geometry. To estimate the distance $M$1 between two vertices, the method constructs a ring-shaped set centered orthogonally to the geodesic from $M$2 to $M$3. By averaging connection statistics over such an orthogonal “ring,” the error due to sample spacing transversally is quadratic rather than linear: bias terms scale as $M$4 rather than $M$5, where $M$6 is the ring thickness. Since the sample complexity for a slab of co-dimension one is $M$7, the optimal achievable scale is $M$8, yielding an error bound of $M$9.

Key geometric insight: For two points $p$0 distance $p$1 apart, auxiliary points $p$2 are chosen in a thin annular region perpendicular to the $p$3 direction. Law of cosines and triangle comparison theorems (and the structure of Riemannian manifolds) ensure that statistical averages over such rings robustly “pin down” the target distance, as transverse fluctuations produce only quadratic error terms.

From Local to Global: Extension and Robustness

• The algorithm bootstraps from coarse, nearly metric-compatible clusters, which can be extracted with weak minimal requirements; these enable estimation of distances at arbitrary scales, supporting “ring” construction.
• Local distance estimates are patched globally by shortest-path (Dijkstra or analogous) procedures, with error propagation well controlled due to the error scaling.
• In the unknown $p$4 (unknown ruler) case, as in [FMR25], the only impossible aspect is absolute scaling. The estimator recovers distances up to a global unknown factor, which is provably optimal.

Rigorous Gromov–Wasserstein Stability

By showing that all pairwise distances are uniformly estimated at error $p$5, and applying sharp inequalities for empirical Wasserstein convergence, the authors prove that the entire metric measure structure (i.e., the induced metric space equipped with the sampling measure) is reconstructed at the optimal rate.

Probabilistic and Geometric Toolset

• Uniform lower bounds on sample occupancy in small annuli (“rings”) are established via covering-number estimates and concentration inequalities.
• The analysis hinges on triangle comparison theorems, regularity lemmas, and Riemannian exponential coordinate estimates for carefully bounding geometric distortion in the orthogonal direction and triangle side lengths.
• The construction is resilient to sample noise, sparsity, unknown link functions, and is effective for general Riemannian manifolds possessing injectivity radius and bounded sectional curvature.

Context in the Literature

Prior works [HJM24, HJM25, FMR25] resolved the problem up to the volumetric barrier, often by using bipartite neighborhood averaging and clustering at scale $p$6. Those methods either require knowledge of the link function or fail at sparse regimes, and none achieve error strictly better than the information-theoretic threshold suggested by empirical measure concentration.

ORDER decisively improves upon this by leveraging the geometric structure not only locally, but in how information is redundantly encoded in annular, transverse regions—the insight being that, for dimensions $p$7, the extra two “quadratic” degrees of freedom encoded by the orthogonal rings overcome the curse of dimensionality imposed by simple sample spacing in the volumetric regime.

Numerical Implications and Strong Claims

• For dimension $p$8, error exponent $p$9 strictly dominates $n^{-1/d}$0, allowing for meaningful improvements in high-dimensional graph-based manifold learning and geometry recovery.
• The Gromov–Wasserstein bound guarantees that, for downstream inference and other geometric machine learning tasks, access to an accurate pairwise distance matrix is unnecessary; a (possibly sparse, possibly noisy, possibly with unknown $n^{-1/d}$1) geometric graph suffices for statistically optimal reconstruction.

Future Directions and Theoretical Implications

The results expose the limitations of naive local averaging in metric learning, demonstrating the continued value of exploiting higher-order geometric dependencies. The authors’ geometric/probabilistic approach is modular, robust to noise, and compatible with more general settings (e.g., weighted graphs, boundary, non-uniform measures). They open the path for adapting similar methods in higher-order random structures (e.g., in optimal transport, embedding theory) and for extending the regime of efficient denoising in manifold learning.

Potential advances may consider sharper minimax optimality, quantitative constants, further relaxations in regularity assumptions, and interactions with more refined classes of random geometric graphs.

Conclusion

This work makes significant technical advances in learning latent Riemannian geometric information from random graph data, breaking a statistical–geometric lower bound (the volumetric barrier) that constrains all previous approaches. By constructing and analyzing ring-shaped, orthogonally oriented clusters for connection statistics, the authors obtain superior error rates, and match the optimal measure convergence in Gromov–Wasserstein distance. The theory is robust, broad in scope, and foundational for fast geometric learning algorithms in high dimensions.