Unique Nearest Point Research Articles

Testing the manifold hypothesis

The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning. The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for fitting a manifold to an unknown probability distribution supported in a separable Hilbert space, only using i.i.d samples from that distribution. More precisely, our setting is the following. Suppose that data are drawn independently at random from a probability distribution $P$ supported on the unit ball of a separable Hilbert space $H$. Let $G(d, V, \tau)$ be the set of submanifolds of the unit ball of $H$ whose volume is at most $V$ and reach (which is the supremum of all $r$ such that any point at a distance less than $r$ has a unique nearest point on the manifold) is at least $\tau$. Let $L(M, P)$ denote mean-squared distance of a random point from the probability distribution $P$ to $M$. We obtain an algorithm that tests the manifold hypothesis in the following sense. The algorithm takes i.i.d random samples from $P$ as input, and determines which of the following two is true (at least one must be): (a) There exists $M \in G(d, CV, \frac{\tau}{C})$ such that $L(M, P) \leq C \epsilon.$ (b) There exists no $M \in G(d, V/C, C\tau)$ such that $L(M, P) \leq \frac{\epsilon}{C}.$ The answer is correct with probability at least $1-\delta$.

Bregman distances and Chebyshev sets

A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given.

Proximinality in geodesic spaces

LetXbe a complete CAT(0)space with the geodesic extension property and Alexandrov curvature bounded below. It is shown that ifCis a closed subset ofX,then the set of points ofXwhich have a unique nearest point inCisGδand of the second Baire category inX.If, in addition,Cis bounded, then the set of points ofXwhich have a unique farthest point inCis dense inX.A proximity result for set-valued mappings is also included.

Closure in a Hilbert space of a prehilbert space Chebyshev set

Let E denote the real inner product space that is the union of all finite dimensional Euclidean spaces. There is a bounded nonconvex set S, that is a subset of E, such that each point of E has a unique nearest point in S. Let H denote the separable Hilbert space that is the completion of space E. A condition is given in order that a point in H have a unique nearest point in the closure of S. We shall also provide an example where the condition fails.

The geometric structure of Chebyshev sets in ℓ∞(n)

A subset M of a normed linear space X is called a Chebyshev set if each x ∈ X has a unique nearest point in M. We characterize Chebyshev sets in l∞(n) in geometric terms and study the approximative properties of sections of Chebyshev sets, suns, and strict suns in l∞(n) by coordinate subspaces.

Chebyshev foam

There is a set F, in a real inner product space E which is the union of all finite-dimensional Euclidean spaces, such that the complement of F has infinitely many components with diameters tending to zero, the union of the components is dense in the space, and each point of E has a unique nearest point in F.

On nearest points in closed convex sets

Main result: For each closed convex subset C of a Banach space X and each there exists an equivalent norm on X for which x has a unique nearest point in C.