We prove that the absolute extendability constant of a finite metric space may be determined by computing relative projection constants certain Lipschitz-free spaces. As an application, we show $\mbox{ae}(3)=4/3$ and $\mbox{ae}(4)\geq (5+4\sqrt{2})/7$. Moreover, discuss how to compute solving linear programming problems.

We study distance-based classification of human actions and introduce a new metric learning approach based on logistic discrimination for the determination of a low-dimensional feature space of increased discrimination power. We argue that for effective distance-based classification, both the optimal projection space and the optimal class representation should be determined. We qualitatively an...

In Computer Vision, the term epipolar geometry describes the intrinsic geometry between two pinhole cameras. While the same model applies to X-ray source and detector, the imaging process itself is very different from visible light. This paper illustrates the epipolar geometry for transmission imaging and makes the connection to Grangeat’s theorem, establishing constraints on redundant projecti...

We develop a supervised dimensionality reduction method, called Lorentzian discriminant projection (LDP), for feature extraction and classification. Our method represents the structures of sample data by a manifold, which is furnished with a Lorentzian metric tensor. Different from classic discriminant analysis techniques, LDP uses distances from points to their within-class neighbors and globa...

Although distance metric learning has been successfully applied to many real-world applications, learning a distance metric from large-scale and high-dimensional data remains a challenging problem. Due to the PSD constraint, the computational complexity of previous algorithms per iteration is at least O(d) where d is the dimensionality of the data. In this paper, we develop an efficient stochas...

To find internal characterizations of certain images of metric spaces is one of central problems in general topology. Arhangel’skiı̆ [1] showed that a space is an open compact image of a metric space if and only if it has a development consisting of point-finite open covers, and some characterizations for certain quotient compact images of metric spaces are obtained in [3, 5, 8]. Recently, Xia [...

A simple geometric procedure is proposed for constructing Wick symbols on cotangent bundles to Riemannian manifolds. The main ingredient of the construction is a method of endowing the cotangent bundle with a formal Kähler structure. The formality means that the metric is lifted from the Riemannian manifold Q to its phase space T ∗Q in the form of formal power series in momenta with the coeffic...

Data analysis is critical to many (if not a ll) Homeland Security missions. Data to be fielded in this domain is typically immense in cardinality (number of records) and immense in dimensionality (features per record). Random Projections (RP) have been proposed as an effective technique for embedding a metric space of dimension d into one of dimension k while retaining bounds on the distortion ...

This paper develops a supervised dimensionality reduction method, Lorentzian Discriminant Projection (LDP), for discriminant analysis and classification. Our method represents the structures of sample data by a manifold, which is furnished with a Lorentzian metric tensor. Different from classic discriminant analysis techniques, LDP uses distances from points to their within-class neighbors and ...

We consider three well-studied polyhedral relaxations for the maximum cut problem: the metric polytope of the complete graph, the metric polytope of a general graph, and the relaxation of the bipartite subgraph polytope. The metric polytope of the complete graph can be described with a polynomial number of inequalities, while the latter two may require exponentially many constraints. We give an...