Geometric Integration for Neural Control Variates
This work addresses a specific bottleneck in variance reduction for Monte Carlo methods, offering an incremental improvement for computational graphics and simulation domains.
The paper tackled the challenge of analytically integrating neural networks for use as control variates in Monte Carlo integration, proposing a method based on domain subdivision and computational geometry in 2D, and demonstrated its application in light transport simulation.
Control variates are a variance-reduction technique for Monte Carlo integration. The principle involves approximating the integrand by a function that can be analytically integrated, and integrating using the Monte Carlo method only the residual difference between the integrand and the approximation, to obtain an unbiased estimate. Neural networks are universal approximators that could potentially be used as a control variate. However, the challenge lies in the analytic integration, which is not possible in general. In this manuscript, we study one of the simplest neural network models, the multilayered perceptron (MLP) with continuous piecewise linear activation functions, and its possible analytic integration. We propose an integration method based on integration domain subdivision, employing techniques from computational geometry to solve this problem in 2D. We demonstrate that an MLP can be used as a control variate in combination with our integration method, showing applications in the light transport simulation.