Estimation of Conformal Metrics

Abstract: We study deformations of the geodesic distances on a domain of R N induced by a function called conformal factor. We show that under a positive reach assumption on the domain (not necessarily a submanifold) and mild assumptions on the conformal factor, geodesics for the conformal metric have good regularity properties in the form of a lower bounded reach. This regularity allows for efficient estimation of the conformal metric from a random point cloud with a relative error proportional to the Hausdorff distance between the point cloud and the original domain. We then establish convergence rates of order n^-1/d that are close to sharp when the intrinsic dimension d of the domain is large, for an estimator that can be computed in O(n^2 ) time. Finally, this paper includes a useful equivalence result between ball graphs and nearest-neighbors graphs when assuming Ahlfors regularity of the sampling measure, allowing to transpose results from one setting to another.

• The paper proves that positive reach of the domain and a Lipschitz, positive conformal factor guarantee uniformly regular conformal geodesics with reach at least min(τ_M/2, f_min/(8κ)).
• The paper develops weighted ball-graph and nearest-neighbor estimators whose relative error reaches O((log n/n)^{1/d}) with sufficiently accurate edge quadrature, while sample-only evaluations yield O((log n/n)^{2/(3d)}).
• The paper extends guarantees beyond smooth manifolds to arbitrary positive-reach sets and establishes a minimax lower bound of n^{-1/(d−1/2)}, leaving the exact optimal rate under minimal assumptions open.

Overview

The paper studies conformal metrics over subsets of Euclidean space: given a closed path-connected domain $M \subset \mathbb{R}^N$ and a positive conformal factor $f: M \to \mathbb{R}_+^*$, the conformal distance between two points is the infimum over Lipschitz paths of the weighted length $\int f(\gamma)\|\dot\gamma\|\,dt$. The author's goal is twofold: first, to establish regularity properties of geodesics for such metrics under a positive reach assumption on $M$ (with no requirement that $M$ be a submanifold), and second, to derive convergence rates for estimating $D_{M,f}$ from an i.i.d. point cloud. The motivating application is the Fermat distance, where $f = \rho^{-\beta}$ for a density $\rho$, used in topological data analysis and metric learning.

Throughout, the sole structural assumptions are that $M$ has positive reach $\tau_M > 0$ (in the sense of Federer) and that $f: M \to \mathbb{R}_+^*$0 is $f: M \to \mathbb{R}_+^*$1-Lipschitz and lower bounded by $f: M \to \mathbb{R}_+^*$2. By McShane extension, $f: M \to \mathbb{R}_+^*$3 may be taken to be defined on all of $f: M \to \mathbb{R}_+^*$4 without loss of generality, which is exploited in constructing practical estimators.

Regularity of Conformal Geodesics

The first main contribution is a lower bound on the reach of geodesics for the conformal metric. Building on the characterization of reach via metric distortion (2602.16466), the paper defines the conformal reach $f: M \to \mathbb{R}_+^*$5 as the infimum of the reaches of all conformal geodesics, and proves it satisfies an analogous arcsine-type characterization. The central result of this section states:

$f: M \to \mathbb{R}_+^*$6

Consequently, every conformal geodesic is a $f: M \to \mathbb{R}_+^*$7 curve whose angular velocity is controlled by its reach. This yields a technical but crucial lemma bounding the deviation of short secant directions from the tangent direction, and the difference of normalized steps on either side of a point, both proportional to step size over reach. These estimates are what later allow the Hausdorff-error term in the approximation bound to be quadratic rather than linear.

Polygonal Approximation on Weighted Graphs

Given a finite point cloud $f: M \to \mathbb{R}_+^*$8, distances are estimated by shortest paths on a graph over $f: M \to \mathbb{R}_+^*$9: either an $\int f(\gamma)\|\dot\gamma\|\,dt$0-ball graph or a $\int f(\gamma)\|\dot\gamma\|\,dt$1-nearest-neighbors graph, with edges weighted by a quadrature formula $\int f(\gamma)\|\dot\gamma\|\,dt$2 that approximates the integral of $\int f(\gamma)\|\dot\gamma\|\,dt$3 along straight segments using $\int f(\gamma)\|\dot\gamma\|\,dt$4 evaluations of $\int f(\gamma)\|\dot\gamma\|\,dt$5 (resolution $\int f(\gamma)\|\dot\gamma\|\,dt$6 corresponds to the trapezoidal weight $\int f(\gamma)\|\dot\gamma\|\,dt$7; $\int f(\gamma)\|\dot\gamma\|\,dt$8 to the exact integral).

For endpoints within distance $\int f(\gamma)\|\dot\gamma\|\,dt$9, the relative distortion of the weights is bounded by

$M$0

where the first term vanishes as resolution grows and the second reflects the curvature of geodesics. The main deterministic approximation theorem then gives, assuming $M$1,

$M$2

Balancing terms via $M$3 and $M$4 makes the total error proportional to the Hausdorff distance. Two points deserve emphasis:

• For the induced metric ($M$5), the paper recovers the quadratic error $M$6 previously obtained under stronger manifold assumptions, showing that positive reach alone—no $M$7 structure, not even a manifold—is sufficient.
• If $M$8 can only be evaluated on the sample ($M$9), the error degrades to order $M$0, matching the rate achievable under the weaker "geodesic smoothness" assumption of prior work.

When the true factor $M$1 is replaced by any estimate $M$2 with $M$3, the total error decomposes additively: it is bounded by twice the domain error plus $M$4. Hence the slowest of the two estimation errors dictates the overall rate.

Estimation from Random Samples

With $M$5 drawn i.i.d. from a $M$6-standard measure $M$7 (a one-sided Ahlfors condition ensuring balls carry mass at least $M$8), the Hausdorff distance $M$9 is shown in expectation to be $D_{M,f}$0, where $D_{M,f}$1. Substituting into the approximation theorem yields the following rates for the ball-graph estimator:

Setting	Parameters	Expected relative error
High resolution ($D_{M,f}$2)	$D_{M,f}$3	$D_{M,f}$4
Resolution $D_{M,f}$5	$D_{M,f}$6	$D_{M,f}$7

In particular, for the induced metric, the estimator converges at rate $D_{M,f}$8 (up to logs) for any set of positive reach. Under the stronger assumption that $D_{M,f}$9 is a $f = \rho^{-\beta}$0 submanifold, the minimax optimal rate for the induced metric is known to be $f = \rho^{-\beta}$1; the present result extends the upper bound to the non-smooth regime at the cost of a slower rate.

Ball graphs versus nearest-neighbor graphs

Optimal tuning of $f = \rho^{-\beta}$2 requires knowledge of the intrinsic dimension $f = \rho^{-\beta}$3. To remove this dependency, the paper establishes a high-probability equivalence between graph families under full $f = \rho^{-\beta}$4-Ahlfors regularity of $f = \rho^{-\beta}$5: with probability at least $f = \rho^{-\beta}$6, the edge sets satisfy

$f = \rho^{-\beta}$7

for $f = \rho^{-\beta}$8 scaling like $f = \rho^{-\beta}$9 up to $\rho$0 factors. Choosing $\rho$1 and $\rho$2 then transfers the ball-graph guarantee to the NN graph at the same rate $\rho$3, with the loss measured over $\rho$4 rather than $\rho$5—the author notes the version over all of $\rho$6 holds but is omitted for brevity. Notably, no knowledge of $\rho$7, the Ahlfors constants, $\rho$8, $\rho$9, or $M$0 is needed to run this estimator.

Complexity: building the $M$1-NN graph costs $M$2; evaluating weights costs $M$3; Dijkstra adds $M$4. With the recommended parameters the overall time is $M$5 per pair of query endpoints.

Minimax Lower Bound

Using Le Cam's method with a two-point construction—a cube versus the same cube with an edge carved by a ball of radius exactly $M$6—the paper proves that over the class of $M$7-standard measures supported on sets of reach at least $M$8, the minimax risk for the induced metric satisfies

$M$9

The construction exploits the fact that carving a region of volume $\tau_M > 0$0 (length $\tau_M > 0$1 along the affected direction, $\tau_M > 0$2 transversally, forced by the standardness constraint) distorts a single interpoint distance by a factor of order $\tau_M > 0$3. The resulting lower bound $\tau_M > 0$4 does not match the upper bound $\tau_M > 0$5, although the gap becomes negligible for large $\tau_M > 0$6. The author argues that closing the gap likely requires a different technique, since the reach assumption caps the distortion achievable by perturbing domains near fixed endpoints at order $\tau_M > 0$7, which appears to force the $\tau_M > 0$8 volume argument. The exact minimax rate under only a positive-reach assumption therefore remains open.

Relation to Smooth-Manifold Estimators

Under a $\tau_M > 0$9 assumption ($f: M \to \mathbb{R}_+^*$00), minimax optimality $f: M \to \mathbb{R}_+^*$01 is achieved by reconstructing the manifold (e.g., via tangential Delaunay complexes) at Hausdorff accuracy $f: M \to \mathbb{R}_+^*$02 and reading off the induced metric—but these procedures are not computationally feasible since they involve non-discrete reconstructions. A concrete pipeline would reconstruct the manifold, sample a fine net over it, and apply the polygonal estimator; the results here certify that this achieves the optimal $f: M \to \mathbb{R}_+^*$03 rate, though at the cost of inflating the effective sample size from $f: M \to \mathbb{R}_+^*$04 to roughly $f: M \to \mathbb{R}_+^*$05.

Examples of Conformal Factors

Two instances connect the theory to existing practice:

• Density-based factors: for $f: M \to \mathbb{R}_+^*$06 (Fermat distance), prior estimators based on edge weights $f: M \to \mathbb{R}_+^*$07 lack convergence guarantees. Coupling the polygonal estimator with a kernel density estimator converging at rate $f: M \to \mathbb{R}_+^*$08 in $f: M \to \mathbb{R}_+^*$09 (available when $f: M \to \mathbb{R}_+^*$10 is a $f: M \to \mathbb{R}_+^*$11 submanifold with $f: M \to \mathbb{R}_+^*$12 density) yields, via the additive error decomposition, an estimator of the Fermat distance converging at the full rate $f: M \to \mathbb{R}_+^*$13—at higher computational cost than discrete Fermat distances.
• Distance-to-measure: $f: M \to \mathbb{R}_+^*$14 is $f: M \to \mathbb{R}_+^*$15-Lipschitz and positively bounded away from zero whenever $f: M \to \mathbb{R}_+^*$16 has no atoms, fitting the framework; it admits an estimator converging at rate $f: M \to \mathbb{R}_+^*$17, faster than $f: M \to \mathbb{R}_+^*$18 and thus not the bottleneck.

Limitations and Open Questions

Several caveats qualify the results. The optimal parameter choices in the ball-graph theorem depend explicitly on $f: M \to \mathbb{R}_+^*$19, $f: M \to \mathbb{R}_+^*$20, and $f: M \to \mathbb{R}_+^*$21; while the NN-graph route removes the need to know them in practice, the theoretical constants still require Ahlfors regularity, and the NN-graph guarantee is stated over $f: M \to \mathbb{R}_+^*$22 rather than $f: M \to \mathbb{R}_+^*$23. The high-resolution regime assumes $f: M \to \mathbb{R}_+^*$24 can be evaluated off-sample; when only sample evaluations are possible ($f: M \to \mathbb{R}_+^*$25), the rate degrades to $f: M \to \mathbb{R}_+^*$26. Most substantively, the minimax rate under the positive-reach-only model remains unresolved: the upper bound $f: M \to \mathbb{R}_+^*$27 and the lower bound $f: M \to \mathbb{R}_+^*$28 do not coincide, and the author offers reasons to suspect that Le Cam-type constructions may be inherently limited in this model. Finally, the density-based application inherits the additional smoothness assumptions ($f: M \to \mathbb{R}_+^*$29 submanifold and $f: M \to \mathbb{R}_+^*$30 density) required by the underlying kernel density estimator.

Conclusion

This work establishes that a positive reach assumption on the domain, together with Lipschitz and lower-bound conditions on the conformal factor, suffices for conformal geodesics to have uniformly positive reach, and consequently for the conformal metric to be estimated from i.i.d. samples at rate $f: M \to \mathbb{R}_+^*$31 (up to logarithms) by an $f: M \to \mathbb{R}_+^*$32-time nearest-neighbor graph algorithm requiring no knowledge of intrinsic parameters. The same machinery applies verbatim to the induced metric of arbitrary positive-reach sets, extending prior manifold-based bounds, and provides quantitative convergence guarantees for Fermat-type metrics where none were previously available.