Accelerating Gaussian Diffusions

Let .(x) be a given probability density proportional to exp(.U(x)) in a high-dimensional Euclidean space Rm. The diffusion dX(t)=..U(X(t))dt+.2dW(t) is often used to sample from .. Instead of ..U(x), we consider diffusions with smooth drift b(x) and having equilibrium .(x). First we study some general properties and then concentrate on the Gaussian case, namely, ..U(x)=Dx with a strictly negative-definite real matrix D and b(x)=Bx with a stability matrix B; that is, the real parts of the eigenvalues of B are strictly negative. Using the rate of convergence of the covariance of X(t) [or together with EX(t)] as the criterion, we prove that, among all such b(x), the drift Dx is the worst choice and that improvement can be made if and only if the eigenvalues of D are not identical. In fact, the convergence rate of the covariance is exp(2.M(B)t), where .M(B) is the maximum of the real parts of the eigenvalues of B and the infimum of .M(B) over all such B is 1/mtrD. If, for example, a "circulant" drift (.U.xm..U.x2,.U.x1..U.x3,.,.U.xm.1..U.x1) is added to Dx, then for essentially all D, the diffusion with this modified drift has a better convergence rate.