DANIEL FREEDMAN

Solving high-dimensional integral equations is a core challenge in science and engineering, primarily due to the curse of dimensionality. Classical numerical solvers, which rely on discretizing the problem domain, suffer from computational costs that grow exponentially with dimension. Monte Carlo methods, often used in machine learning, sidestep this scaling by providing convergence rates independent of dimension. However, they require a large number of samples to accurately approximate integrals. We show that naive sampling can bias the loss function, resulting in inaccurate solutions, especially in high-dimensional settings. We introduce Low Rank Polynomials Sum (LoRPS), a method that can scale to high dimension and solve the challenge of accurate integral estimation. LoRPS leverages a low-rank, separable polynomial structure that allows the integrals in the loss function to be computed analytically when the terms of the equation are also polynomial, avoiding any sampling-induced error while maintaining minimal computational overhead. For analytic target functions, we prove that LoRPS becomes more accurate at an asymptotic rate that is independent of dimension as its representation capacity increases, with the effects of dimension and smoothness reflected in problem-dependent constants. On challenging high-dimensional benchmarks of the appropriate smoothness, it consistently achieves higher accuracy while using at least 3.5x less memory than existing methods.