Hariharan Narayanan

Random Walks on Polytopes and an Affine Interior Point Method for Linear Programming

Let K be a polytope in Rn defined by m linear inequalities. We give a new Markov chain algorithm to draw a nearly uniform sample from K. The underlying Markov chain is the first to have a mixing time that is strongly polynomial when started from a “central” point. We use this result to design an affine interior point algorithm that does a single random walk to solve linear programs approximately.

Heat Flow and a Faster Algorithm to Compute the Surface Area of a Convex Body

We draw on the observation that the amount of heat diffusing outside of a heated body in a short period of time is proportional to its surface area, to design a simple algorithm for approximating the surface area of a convex body given by a membership oracle. Our method has a complexity of O*(n4), where n is the dimension, compared to O*( n8.5) for the previous best algorithm. We show that our complexity cannot be improved given the current state-of-the-art in volume estimation

Heat flow and a faster algorithm to compute the surface area of a convex body

We draw on the observation that the amount of heat diffusing outside of a heated body in a short period of time is proportional to its surface area, to design a simple algorithm for approximating the surface area of a convex body given by a membership oracle. Our method has a complexity of O*(n4), where n is the dimension, compared to O*(n8) for the previous best algorithm. We show that our complexity cannot be improved given the current state-of-the-art in volume estimation.

Sampling Hypersurfaces through Diffusion

We are interested in efficient algorithms for generating random samples from geometric objects such as Riemannian manifolds. As a step in this direction, we consider the problem of generating random samples from smooth hypersurfaces that may be represented as the boundary

Distributed averaging in the presence of a sparse cut

\partial A

Sample Complexity of Testing the Manifold Hypothesis

of a domain Ai¾? i¾?dof Euclidean space. Ais specified through a membership oracle and we assume access to a blackbox that can generate uniform random samples from A. By simulating a diffusion process with a suitably chosen time constant t, we are able to construct algorithms that can generate points (approximately) on

On the Relation Between Low Density Separation, Spectral Clustering and Graph Cuts

\partial A

Geometric complexity theory III: on deciding nonvanishing of a Littlewood---Richardson coefficient

according to a (approximately) uniform distribution. We have two classes of related but distinct results. First, we consider Ato be a convex body whose boundary is the union of finitely many smooth pieces, and provide an algorithm (Csample) that generates (almost) uniformly random points from the surface of this body, and prove that its complexity is

Escaping the Local Minima via Simulated Annealing: Optimization of Approximately Convex Functions

O^*(\frac{d^4}{\epsilon})

Random Walk Approach to Regret Minimization

per sample, where i¾?is the variation distance. Next, we consider Ato be a potentially non-convex body whose boundary is a smooth (co-dimension one) manifold with a bound on its absolute curvature and diameter. We provide an algorithm (Msample) that generates almost uniformly random points from