Related papers: Universally consistent estimation of the reach
We study the estimation of the reach, an ubiquitous regularity parameter in manifold estimation and geometric data analysis. Given an i.i.d. sample over an unknown $d$-dimensional $\mathcal{C}^k$-smooth submanifold of $\mathbb{R}^D$, we…
Motivated by questions of manifold learning, we study a sequence of random manifolds, generated by embedding a fixed, compact manifold $M$ into Euclidean spheres of increasing dimension via a sequence of Gaussian mappings. One of the…
Various problems in manifold estimation make use of a quantity called the reach, denoted by $\tau\_M$, which is a measure of the regularity of the manifold. This paper is the first investigation into the problem of how to estimate the…
The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of…
This paper contains three main results. Firstly, we give an elementary proof of the following statement: Let $M$ be a (closed, in both the geometrical and topological sense of the word) topological manifold embedded in $\mathbb{R}^d$. If…
The reach of a submanifold of $\mathbb{R}^N$ is defined as the largest radius of a tubular neighbourhood around the submanifold that avoids self-intersections. While essential in geometric and topological applications, computing the reach…
A subset of a topological space is said to be \emph{universally measurable} if it is measured by the completion of each countably additive $\sigma$-finite Borel measure on the space, and \emph{universally null} if it has measure zero for…
It is well-known since the seminal work of Herbert Federer [Trans. of the AMS, 1959] that submanifolds of class $C^{1,1}$ have positive reach. In this paper, we extend this property to less regular submanifolds by using the notion of…
The behavior of the volume of the tube of distance r, around a given compact set M, is an old and important question with relations to many fields, like differential geometry, geometric measure theory, integral geometry, and also…
Determining the reachable set for a given nonlinear system is critically important for autonomous trajectory planning for reach-avoid applications and safety critical scenarios. Providing the reachable set is generally impossible when the…
The notion of relative universality with respect to a {\sigma}-field was introduced to establish the unbiasedness and Fisher consistency of an estimator in nonlinear sufficient dimension reduction. However, there is a gap in the proof of…
The convexity of a set can be generalized to the two weaker notions of reach and $r$-convexity; both describe the regularity of a set's boundary. For any compact subset of $\mathbb{R}^d$, we provide methods for computing upper bounds on…
The general aim of manifold estimation is reconstructing, by statistical methods, an $m$-dimensional compact manifold $S$ on ${\mathbb R}^d$ (with $m\leq d$) or estimating some relevant quantities related to the geometric properties of $S$.…
The study of shape restrictions of subsets of $\mathbb{R}^d$ have several applications in many areas, being convexity, $r$-convexity, and positive reach, some of the most famous, and typically imposed in set estimation. The following…
Assumptions on the reach are crucial for ensuring the correctness of many geometric and topological algorithms, including triangulation, manifold reconstruction and learning, homotopy reconstruction, and methods for estimating curvature or…
We give a complete characterization of compact sets with positive reach (=proximally $C^1$ sets) in the plane and of one-dimensional sets with positive reach in ${\mathbb R}^d$. Further, we prove that if $\emptyset \neq A\subset{\mathbb…
This article deals with the existence of hypersurfaces minimizing general shape functionals under certain geometric constraints. We consider as admissible shapes orientable hypersurfaces satisfying a so-called reach condition, also known as…
Motivated by set estimation problems, we consider three closely related shape conditions for compact sets: positive reach, r-convexity and rolling condition. First, the relations between these shape conditions are analyzed. Second, we…
We propose two novel approaches to address a critical problem of reach measurement across multiple media -- how to estimate the reach of an unobserved subset of buying groups (BGs) based on the observed reach of other subsets of BGs.…
Federer's characterization of sets of finite perimeter states (in Euclidean spaces) that a set is of finite perimeter if and only if the measure-theoretic boundary of the set has finite Hausdorff measure of codimension one. In complete…