Algorithmic aspects of convexity . Santosh Vempala ’ s lecture

Testing Geometric Convexity

We consider the problem of determining whether a given set S in R is approximately convex, i.e., if there is a convex set K ∈ R such that the volume of their symmetric difference is at most vol(S) for some given . When the set is presented only by a membership oracle and a random oracle, we show that the problem can be solved with high probability using poly(n)(c/ ) oracle calls and computation...

Recent Progress and Open Problems in Algorithmic Convex Geometry

This article is a survey of developments in algorithmic convex geometry over the past decade. These include algorithms for sampling, optimization, integration, rounding and learning, as well as mathematical tools such as isoperimetric and concentration inequalities. Several open problems and conjectures are discussed on the way.

Algorithms for Data Mining

Data of massive size are now available in a wide variety of fields and come with great promise. In theory, these massive data sets allow data mining and exploration on a scale previously unimaginable. However, in practice, it can be difficult to apply classic data mining techniques to such massive data sets due to their sheer size. In this thesis, we study three algorithmic problems in data min...

Geometri algorithms for online optimization

Geometric Random Walks: A Survey∗

The developing theory of geometric random walks is outlined here. Three aspects — general methods for estimating convergence (the “mixing” rate), isoperimetric inequalities in Rn and their intimate connection to random walks, and algorithms for fundamental problems (volume computation and convex optimization) that are based on sampling by random walks — are discussed.