The Sliced-Wasserstein (SW) distance between probability measures is defined as the average of the Wasserstein distances resulting for the associated one-dimensional projections. As a consequence, the SW distance can be written as an integral with respect to the uniform measure on the sphere and the Monte Carlo framework can be employed for calculating the SW distance. Spherical harmonics are polynomials on the sphere that form an orthonormal basis of the set of square-integrable functions on the sphere. Putting these two facts together, a new Monte Carlo method, hereby referred to as Spherical Harmonics Control Variates (SHCV), is proposed for approximating the SW distance using spherical harmonics as control variates. The resulting approach is shown to have good theoretical properties, e.g., a no-error property for Gaussian measures under a certain form of linear dependency between the variables. Moreover, an improved rate of convergence, compared to Monte Carlo, is established for general measures. The convergence analysis relies on the Lipschitz property associated to the SW integrand. Several numerical experiments demonstrate the superior performance of SHCV against state-of-the-art methods for SW distance computation.

Driven by several successful applications such as in stochastic gradient descent or in Bayesian computation, control variates have become a major tool for Monte Carlo integration. However, standard methods do not allow the distribution of the particles to evolve during the algorithm, as is the case in sequential simulation methods. Within the standard adaptive importance sampling framework, a simple weighted least squares approach is proposed to improve the procedure with control variates. The procedure takes the form of a quadrature rule with adapted quadrature weights to reflect the information brought in by the control variates. The quadrature points and weights do not depend on the integrand, a computational advantage in case of multiple integrands. Moreover, the target density needs to be known only up to a multiplicative constant. Our main result is a non-asymptotic bound on the probabilistic error of the procedure. The bound proves that for improving the estimate's accuracy, the benefits from adaptive importance sampling and control variates can be combined. The good behavior of the method is illustrated empirically on synthetic examples and real-world data for Bayesian linear regression.

The impact of within-die transistor variability has increased with CMOS technology scaling up to the point where it has emerged as a systematic problem for the designer. Estimating extremely low failure rate, i.e. “high-sigma” probabilities, by the conventional Monte Carlo (MC) approach requires millions of simulation runs, making it an impractical approach for circuit designers. To overcome this problem, alternative failure estimation methodologies, which require a smaller number of runs have been proposed. In this paper, we propose a novel methodology called “gradient importance sampling” (GIS) for fast statistical extraction of high-sigma circuit characteristics. It is based on conventional Importance Sampling combined with a gradient-based approach to find the most probable failure point (MPFP). By applying GIS to extract SRAM dynamic characteristics in 28nm FDSOI CMOS, we show that the proposed methodology is straightfor-ward, computationally efficient and the results are in line with those obtained via standard MC. To the best of our knowledge, the GIS results are the best in their class for low failure rate estimation.