Heat-Kernel Entropy Profiles and Geometric Effective Sample Size for Weighted Measures on Manifolds
arXiv:2607.06696v1 Announce Type: cross Abstract: Weighted empirical measures on compact manifolds arise in importance sampling, particle approximations, posterior summaries, quadrature, and representation learning. Standard weight-only summaries, such as ordinary effective sample size, ignore the geometry of the support. We introduce heat-kernel entropy profiles, a multiscale summary that diffuses weighted atoms by intrinsic heat flow and tracks nonuniformity across scales. For order-two R\'enyi entropy, the profile is computable from pairwise heat-kernel overlaps and yields a geometric effec
This research addresses fundamental limitations in current statistical methods for evaluating weighted empirical measures, which are increasingly important across various AI and data science applications.
Improved methods for evaluating weighted measures on complex manifolds can lead to more robust and geometrically aware algorithms in machine learning, affecting fields from representation learning to uncertainty quantification.
This introduces a novel multiscale summary, heat-kernel entropy profiles, which incorporates geometric context into effective sample size calculations, moving beyond simpler weight-only metrics.
- · AI/ML researchers
- · Data scientists
- · Optimization algorithm developers
- · Robotics and autonomous systems
- · Developers relying solely on traditional weight-only metrics
- · Applications with high sensitivity to geometric inaccuracy
More accurate and reliable assessment of sample quality in complex data distributions.
Development of new algorithms that explicitly leverage geometric effective sample size for improved performance in sampling and learning tasks.
Potentially enables more efficient and robust generative models and reinforcement learning applications where manifold geometry is critical.
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Read at arXiv cs.LG