We show that the Hausdorff metric over constant-size pointsets in constant-dimensional Euclidean space admits an embedding into constant-dimensional `∞ space with constant distortion. More specifically for any s, d ≥ 1, we obtain an embedding of the Hausdorff metric over pointsets of size s in d-dimensional Euclidean space, into `s ∞ with distortion sO(s+d). We remark that any metric space M ad...

The paper proposes a new method to perform foreground detection by means of background modeling using the tensor concept. Sometimes, statistical modelling directly on image values is not enough to achieve a good discrimination. Thus the image may be converted into a more information rich form, such as a tensor field, to yield latent discriminating features. Taking into account the theoretically...

We show that for every α > 0, there exist n-point metric spaces (X, d) where every “scale” admits a Euclidean embedding with distortion at most α, but the whole space requires distortion at least Ω( √ α log n). This shows that the scale-gluing lemma [Lee, SODA 2005] is tight, and disproves a conjecture stated there. This matching upper bound was known to be tight at both endpoints, i.e. when α ...

We show that for every n-point metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight w(T ) = O(k · n) · w(MST (M)), and a spanning tree T ′ with weight w(T ′) = O(k) · w(MST (M)) and unweighted diameter O(k · n). These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently...

We study the problem of representing all distances between $n$ points in ${\mathbb R}^d$, with arbitrarily small distortion, using as few bits possible. give asymptotically tight bounds for this problem, Euclidean metrics, $\ell_1$ (also known Manhattan)-metrics, and general metrics. Our metrics mark first improvement over compression schemes based on discretizing classical dimensionality reduc...

In this paper, we consider two important problems defined on finite metric spaces, and provide efficient new algorithms and approximation schemes for these problems on inputs given as graph shortest path metrics or high-dimensional Euclidean metrics. The first of these problems is the greedy permutation (or farthest-first traversal) of a finite metric space: a permutation of the points of the s...

Several Riemannian metrics and families of were defined on the manifold Symmetric Positive Definite (SPD) matrices. Firstly, we formalize a common general process to define metrics: principle deformed metrics. We relate recently introduced family alpha-Procrustes class mean kernel by providing sufficient condition under which elements former belong latter. Secondly, focus balanced bilinear form...

Most existing appearance models for visual tracking usually construct a pixel-based representation of object appearance so that they are incapable of fully capturing both global and local spatial layout information of object appearance. In order to address this problem, we propose a novel spatial LogEuclidean appearance model (referred as SLAM) under the recently introduced Log-Euclidean Rieman...

We show that greedy geometric routing schemes exist for the Euclidean metric in R, for 3-connected planar graphs, with coordinates that can be represented succinctly, that is, with O(log n) bits, where n is the number of vertices in the graph.

For a set P of n points in R, the Euclidean 2-center problem computes a pair of congruent disks of the minimal radius that cover P . We extend this to the (2, k)-center problem where we compute the minimal radius pair of congruent disks to cover n − k points of P . We present a randomized algorithm with O(nk log n) expected running time for the (2, k)-center problem. We also study the (p, k)-ce...