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This research introduces a Gauss-Hermite quadrature method to approximate differential entropy for Gaussian mixtures, addressing the lack of closed-form solutions in multimodal uncertainty modeling. The authors further propose a Hermite polynomial surrogate for action-space optimization, demonstrating superior accuracy and reduced regret compared to Taylor approximations in radar pointing benchmarks. These advances offer more robust numerical tools for handling complex probability distributions and continuous control problems in machine learning systems.
Gauss--Hermite Quadrature for Gaussian-Mixture Entropy with an Action-Space Hermite Surrogate
arXiv:2608.21467v1 Announce Type: cross Abstract: Gaussian distributions are used to model uncertainty in signals and states, and Gaussian mixtures are often used when the underlying distribution is multimodal. Unlike a single Gaussian, a Gaussian mixture generally has no closed-form expression for differential entropy and therefore requires numerical approximation. We propose a Gauss--Hermite quadrature method for evaluating Gaussian mixture differential entropy. The quadrature order controls the numerical resolution of the approximation. The method is evaluated on one- and two-dimensional Gaussian mixture benchmarks against Taylor approximations, analytic entropy bounds, and numerical integration references. For repeated optimization over continuous actions, we also propose a Hermite polynomial surrogate in action space. In a radar pointing benchmark, its second-order form achieves substantially lower surrogate error and optimizer regret than a second-order Taylor surrogate based on local derivatives at the nominal action, while both methods use nine direct objective evaluations per replanning step. The Hermite surrogate also improves pointing performance in the tested benchmark.
Abstract:Gaussian distributions are used to model uncertainty in signals and states, and Gaussian mixtures are often used when the underlying distribution is multimodal. Unlike a single Gaussian, a Gaussian mixture generally has no closed-form expression for differential entropy and therefore requires numerical approximation. We propose a Gauss--Hermite quadrature method for evaluating Gaussian mixture differential entropy. The quadrature order controls the numerical resolution of the approximation. The method is evaluated on one- and two-dimensional Gaussian mixture benchmarks against Taylor approximations, analytic entropy bounds, and numerical integration references. For repeated optimization over continuous actions, we also propose a Hermite polynomial surrogate in action space. In a radar pointing benchmark, its second-order form achieves substantially lower surrogate error and optimizer regret than a second-order Taylor surrogate based on local derivatives at the nominal action, while both methods use nine direct objective evaluations per replanning step. The Hermite surrogate also improves pointing performance in the tested benchmark.
Machine Learning (stat.ML); Information Theory (cs.IT); Machine Learning (cs.LG)
arXiv:2608.21467 [stat.ML]
(or arXiv:2608.21467v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2608.21467 Focus to learn more arXiv-issued DOI via DataCite
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