k-Means: Outliers-Resistant Clustering+++

The k-means problem is to compute a set of k centers (points) that minimizes the sum of squared distances to a given set of n points in a metric space. Arguably, the most common algorithm to solve it is k-means++ which is easy to implement and provides a provably small approximation error in time that is linear in n. We generalize k-means++ to support outliers in two sense (simultaneously): (i) nonmetric spaces, e.g., M-estimators, where the distance dist(p,x) between a point p and a center x is replaced by mindist(p,x),c for an appropriate constant c that may depend on the scale of the input. (ii) k-means clustering with m≥1 outliers, i.e., where the m farthest points from any given k centers are excluded from the total sum of distances. This is by using a simple reduction to the (k+m)-means clustering (with no outliers).

Farthest points on most Alexandrov surfaces

We study global maxima of distance functions on most Alexandrov surfaces with curvature bounded below, where most is used in the sense of Baire categories.

Horosphere slab separation theorems in manifolds without conjugate points

Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

Asymptotic farthest points and extreme points

Let X be a normed space, G a nonempty bounded subset of X and fxng a bounded sequence in X. In this article, we introduce and discuss the concept of asymptotic farthest points of fxng in G, which is a new definition in abstract approximation theory. Then, by applying the topics of functional analysis, we investigate the relation between this new concept and the concepts of extreme points and convexity. In particular, one of the main purposes of this paper is to study conditions under which the existence (uniqueness) of asymptotic farthest point of fxng in G is equivalent to the existence (uniqueness) of asymptotic farthest point of fxng in ext(G) or co(G).

On the farthest points in convex metric spaces and linear metric spaces

We prove some results on the farthest points in convex metric spaces and in linear metric spaces. The continuity of the farthest point map and characterization of strictly convex linear metric spaces in terms of farthest points are also discussed.