Antithetic variates in higher dimensions

We introduce the concept of multidimensional antithetic as the absolute minimum of the covariance defined on the orthogonal group by $A\mapsto Cov(f(\xi ),f(A\xi ))$ where $\xi$ is a standard $N$-dimensional normal random variable and $f:{\mathbb{R}}^N\to \mathbb{R}$ is an almost everywhere differentiable function. The antithetic matrix is designed to optimise the calculation of $E[f(\xi )]$ in a Monte Carlo simulation. We present an iterative annealing algorithm that dynamically incorporates the estimation of the antithetic matrix within the Monte Carlo calculation.