Generalized Discrepancy of Random Points Improves Bounds for High-Dimensional Sampling with Optimal Densities

The distribution of random points in high-dimensional spaces presents a long-standing challenge, often hampered by what is known as the curse of dimensionality. Erich Novak and Friedrich Pillichshammer investigate this problem by focusing on the discrepancy of these point sets, a measure of how evenly they fill a space. Their work clarifies the relationship between the number of points, the dimensionality of the space, and the achievable uniformity, revealing significant improvements through a technique akin to importance sampling. By allowing for non-uniform sampling densities, the researchers demonstrate that optimally chosen point distributions can substantially reduce discrepancy, although the fundamental limitations imposed by dimensionality ultimately persist, particularly when dealing with a small number of points.

This research offers new insights into the behaviour of random point sets and suggests similar constraints likely apply to deterministic arrangements as well. Discrepancy and Tractability in High Dimensions The study investigates the efficiency of numerical integration in high-dimensional spaces, focusing on the interplay between discrepancy, tractability, and weighted integration. Discrepancy measures how uniformly points are distributed, while tractability refers to whether the complexity of a problem grows at a manageable rate with increasing dimensions. Researchers explore how these concepts relate to function spaces and numerical methods like Quasi-Monte Carlo and Monte Carlo techniques. The central question addresses under what conditions high-dimensional integration can be performed efficiently, and what limitations exist for various numerical approaches. The research establishes lower bounds on integration error, demonstrating that certain function classes inherently become intractable in high dimensions due to exponential error growth. Conversely, the team derives upper bounds for specific methods, particularly Quasi-Monte Carlo, showing that polynomial convergence rates are achievable under certain conditions, making the problem tractable. Optimal importance sampling, a variance reduction technique, is also investigated, revealing its potential to significantly reduce integration error for specific problems. This work provides a rigorous theoretical foundation for understanding the complexity of high-dimensional integration, guiding the design of efficient algorithms and offering implications for applications in finance, physics, engineering, and machine learning.,.

Deterministic Point Sets Beat Random Discrepancy Scientists have made significant advances in understanding the discrepancy of random point sets in high dimensions, particularly for small values of p. The research moves beyond the limitations of classical discrepancy analysis by exploring generalized discrepancies obtained through non-uniform sampling densities and corresponding quadrature weights. The core of their approach involves drawing random points independently according to a product density, then evaluating integrands with weights inversely proportional to that density. This technique allows for analysis of the expected generalized discrepancy and demonstrates the existence of deterministic point sets with a discrepancy no larger than this expectation. To achieve improvements, scientists focused on optimizing the product density used for sampling. For the case of p equals 2, they completely solved a variational problem, recovering an optimal density previously found in earlier work and improving the classical bound. Extending this to general values of p, the team determined the optimal one-dimensional density using a nonlinear equation, resulting in an improved exponential factor governing the dependence on dimension.

Results demonstrate that random points drawn from optimally chosen product measures can significantly outperform currently known deterministic point sets regarding generalized discrepancy. However, even with optimal densities, the curse of dimensionality persists for all p greater than or equal to 1, with the exponential growth rate in dimension becoming more severe as p approaches 1. This suggests that the curse of dimensionality also holds for the classical L1-discrepancy with deterministic points, a problem that remains open.

The team’s method provides a framework for analyzing the impact of non-uniform sampling and offers insights into the fundamental limits of high-dimensional integration.,.

Optimal Point Distributions Reduce High-Dimensional Discrepancy Scientists have achieved significant improvements in understanding the discrepancy of random point sets in high dimensions, particularly for small values of p in the Lp-discrepancy measure. This work focuses on generalized discrepancies, allowing for non-uniform sampling densities and corresponding quadrature weights, and reveals how strategically chosen sampling distributions can enhance performance. The research demonstrates that random points drawn from optimally chosen product densities yield substantially improved upper bounds compared to traditional methods using uniformly distributed points. For p equals 2, scientists completely solved the underlying variational problem, recovering an optimal density and improving the classical bound. Extending this analysis to general values of p between 1 and infinity, the team showed that the optimal one-dimensional density is uniquely determined by a nonlinear equation, leading to an improved exponential factor governing the dependence on dimension. Experiments reveal that even with optimal densities, the curse of dimensionality persists for random points when p is greater than or equal to 1, becoming more pronounced as dimensions increase. These findings demonstrate that strategically chosen sampling distributions can significantly outperform currently known deterministic point sets with respect to generalized discrepancy, while also clarifying the limitations imposed by the curse of dimensionality in high-dimensional integration.,.

Optimal Sampling Bounds in High Dimensions This research significantly advances understanding of discrepancy in high-dimensional random point sets, particularly for small discrepancy values where traditional methods struggle due to the curse of dimensionality.

Scientists have developed new upper bounds for generalized discrepancies by employing non-uniform sampling densities and associated quadrature weights, demonstrating that strategically chosen product densities can substantially improve performance. Notably, these bounds are explicit and optimal in certain cases, and provide asymptotic estimates for the general case, representing a form of importance sampling tailored to the underlying function space.

The team’s findings reveal that even with optimal sampling, the curse of dimensionality persists for random points when discrepancy is small, becoming more pronounced as dimensions increase. Furthermore, the study clarifies stability considerations for quadrature formulas within the specific function space under investigation, demonstrating that algorithms achieving small integration errors also exhibit small norms within that space. While acknowledging that algorithms with large weights may be sensitive to noisy data, the researchers suggest mitigating this risk through regularization of the optimal density. Future work will focus on refining asymptotic bounds and exploring the implications of these findings for various applications requiring efficient high-dimensional integration and optimization. 👉 More information 🗞 Generalized Discrepancy of Random Points 🧠 ArXiv: https://arxiv.org/abs/2512.08364 Tags: