LoRPS: Solving High Dimensional Integral Equations With Low-Rank Polynomials Sum

TMLR Paper9040 Authors

Abstract: 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, avoiding any sampling-induced error while maintaining minimal computational overhead. We prove that under mild assumptions, LoRPS mitigates the curse of dimensionality. On challenging high-dimensional benchmarks, it consistently achieves higher accuracy while using at least \(3.5\times\) less memory than existing methods.

Submission Type: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~Arya_Mazumdar1

Submission Number: 9040