Lee-Ad Gottlieb

Searching dynamic point sets in spaces with bounded doubling dimension

We present a new data structure that facilitates approximate nearest neighbor searches on a dynamic set of points in a metric space that has a bounded doubling dimension. Our data structure has linear size and supports insertions and deletions in O(log n) time, and finds a (1+ε)-approximate nearest neighbor in time O(log n) + (1/ε)O(1). The search and update times hide multiplicative factors that depend on the doubling dimension; the space does not. These performance times are independent of the aspect ratio (or spread) of the points.

Strong Stability Preserving Properties of Runge–Kutta Time Discretization Methods for Linear Constant Coefficient Operators

Strong stability preserving (SSP) high order Runge–Kutta time discretizations were developed for use with semi-discrete method of lines approximations of hyperbolic partial differential equations, and have proven useful in many other applications. These high order time discretization methods preserve the strong stability properties of first order explicit Euler time stepping. In this paper we analyze the SSP properties of Runge Kutta methods for the ordinary differential equation ut=Lu where L is a linear operator. We present optimal SSP Runge–Kutta methods as well as a bound on the optimal timestep restriction. Furthermore, we extend the class of SSP Runge–Kutta methods for linear operators to include the case of time dependent boundary conditions, or a time dependent forcing term.

An Optimal Dynamic Spanner for Doubling Metric Spaces

A t-spanner is a graph on a set of points Swith the following property: Between any pair of points there is a path in the spanner whose total length is at most ttimes the actual distance between the points. In this paper, we consider points residing in a metric space equipped with doubling dimension i¾?, and show how to construct a dynamic (1 + i¾?)-spanner with degree i¾?i¾? O(i¾?)in

Efficient Classification for Metric Data

O(\frac{\log n}{\varepsilon^{O(\lambda)}})

A nonlinear approach to dimension reduction

update time. When i¾?and i¾?are taken as constants, the degree and update times are optimal.

Matrix sparsification and the sparse null space problem

Recent advances in large-margin classification of data residing in general metric spaces (rather than Hilbert spaces) enable classification under various natural metrics, such as string edit and earthmover distance. A general framework developed for this purpose left open the questions of computational efficiency and of providing direct bounds on generalization error. We design a new algorithm for classification in general metric spaces, whose runtime and accuracy depend on the doubling dimension of the data points, and can thus achieve superior classification performance in many common scenarios. The algorithmic core of our approach is an approximate (rather than exact) solution to the classical problems of Lipschitz extension and of nearest neighbor search. The algorithms generalization performance is guaranteed via the fat-shattering dimension of Lipschitz classifiers, and we present experimental evidence of its superiority to some common kernel methods. As a by-product, we offer a new perspective on the nearest neighbor classifier, which yields significantly sharper risk asymptotics than the classic analysis.

Efficient Regression in Metric Spaces via Approximate Lipschitz Extension

AbstractThe

A Light Metric Spanner

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