Learning mixtures of arbitrary gaussians
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Sampling s-Concave Functions: The Limit of Convexity Based Isoperimetry
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
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This paper deals with the normal density of n dependent random variables. This is a function of the form: ce(-x/sup T/Ax) where A is an n/spl times/n positive definite matrix, a: is the n-vector of the random variables and c is a suitable constant. The first problem we consider is the (approximate) evaluation of the integral of this function over the positive orthant /spl int/(x/sub 1/=0)/sup /spl infin///spl int/(x/sub 2/=0)/sup /spl infin///spl middot//spl middot//spl middot//spl int/(x/sub n/=0)/sup /spl infin//ce(-x/sup T/Ax). This problem has a long history and a substantial literature. Related to it is the problem of drawing a sample from the positive orthant with probability density (approximately) equal to ce(-x/sup T/Ax). We solve both these problems here in polynomial time using rapidly mixing Markov Chains. For proving rapid convergence of the chains to their stationary distribution, we use a geometric property called the isoperimetric inequality. Such an inequality has been the subject of recent papers for general log-concave functions. We use these techniques, but the main thrust of the paper is to exploit the special property of the normal density to prove a stronger inequality than for general log-concave functions. We actually consider first the problem of drawing a sample according to the normal density with A equal to the identity matrix from a convex set K in R/sup n/ which contains the unit ball. This problem is motivated by the problem of computing the volume of a convex set in a way we explain later. Also, the methods used in the solution of this and the orthant problem are similar.