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.
This paper studies metrics over subsets of the Euclidean space (R N , ∥•∥) obtained by conformal deformation of the shortest-path metric via a positive function. We are particularly interested in the regularity of such metrics and in their estimation via i.i.d. sampling of points.
▶ Definition 1.1. Let M ⊂ R N be a closed path-connected domain and f : M → R * + be a conformal factor. For all x, y ∈ M the conformal distance between x and y over M via f is defined as
where Γ M (x, y) is the set of all Lipschitz paths γ : I → M where I = [a, b] is a nontrivial segment, γ(a) = x and γ(b) = y. The quantity minimized over all paths γ is denoted
and referred to as the conformal length of the curve γ via f . In the case where f = 1, we write D M (x, y) = D M,1 (x, y) and |γ| = I ∥ γ∥, respectively the distance between x and y induced by the ambient metric over M and the Euclidean length of a curve γ.
In the following, to ensure regularity of the conformal metric, the domain M is assumed to have positive reach τ M , which we recall is defined as the supremum of all r > 0 such that any point in the offset M r = {x ∈ R N : d(x, M ) < r} has a unique projection onto M -where d(x, M ) = inf y∈M ∥x -y∥ denotes the distance from point x to subset M -see [11]. Moreover, the conformal factor f is assumed to be κ-Lipschitz and lower bounded by f min > 0. These are the sole assumptions used in this paper regarding M and f . Notice that f is defined over M in Definition 1.1 as paths are constrained to the domain. Since any Lipschitz function defined over a subset of R N can be extended to the whole space preserving its Lipschitz property and lower bound (see [17, Theorem 1] for instance), one may assume without loss of generality that f is defined over the entire space R N , which is useful to estimate the conformal metric efficiently. The connectedness and positive reach of M ensure that for any endpoints x, y ∈ M , there exists a Lipschitz path from x to y in M , so that Γ M (x, y) is always nonempty and D M,f (x, y) is always finite. This can be deduced from Lemma 2.1 below. Most of the time, a path γ is chosen to be parameterized with constant velocity, i.e., ∥ γ∥ is constant over I, with I being either [0, 1] or [0, |γ|]. The latter is referred to as an arc-length parameterization, where ∥ γ∥ = 1 almost everywhere.
The induced metric D M = D M,1 is the metric induced by the ambient space R N onto M . If M is a C 1 submanifold of R N and f is C 1 , D M,f is a Riemannian metric with tensor f (x) 2 • g x at point x ∈ M where g x is the tensor of the induced metric D M at x. The term “conformal” used in this paper is borrowed from the Riemannian literature. Finally, if µ is a measure over a submanifold M with density ρ with regard to (w.r.t.) the volume form of M and f is a negative power of the density ρ, D M,f has already been studied and is sometimes referred to as the Fermat distance [14]. This particular case is one of the main motivations for this article as the Fermat distance has been used for various practical applications, see for instance [13] and the references therein. Particular examples of the conformal factor f are discussed in Section 5.
Conformal metrics are included in the more general class of length-spaces, for which it is known that the infimum in Definition 1.1 is in fact a minimum [7,Theorem 2.5.23]. That is, for all x, y ∈ M there exists a path γ ∈ Γ M (x, y) such that D M,f (x, y) = |γ| f , called a minimizing geodesic-and shortened to geodesic in this paper for brevity. Denote Γ ⋆ M,f (x, y) the set of such geodesics between two endpoints x and y along with Γ ⋆ M,f = x̸ =y∈M Γ ⋆ M,f (x, y) the set of all geodesics w.r.t. the conformal metric. We discuss in Section 2 the regularity of geodesics w.r.t. D M,f and show in Proposition 2.4 that under the aforementioned assumptions on M and f , geodesics have positive reach that is lower bounded by an explicit constant depending on the reach of the domain and on the conformal factor. In particular, any geodesic may be parameterized as a C 1,1 curve with an explicit upper bound on the Lipschitz constant of its first derivative.
We then show in Section 3 that the conformal metric can be approached using polygonal paths on a weighted graph built from a point cloud X ⊂ M , provided that the graph only contains edges of length at most some threshold r and that X is close to M in Hausdorff distance. The small edges condition ensures that paths on the graph cannot venture too far outside the domain. This kind of construction is the same as the one used for the Isomap algorithm [5], although we allow more generality by adapting the weights to the conformal factor f . When f = 1 and M is a geodesically convex C 2 submanifold, [5] provides a relative error bound of order r 2 τ 2 + ρ r where τ is the minimum radius of curvature of M and ρ denotes the Hausdorff distance between M and X. The term depending on ρ can be made quadratic as shown in [2], where an as
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