Searching permutations for constructing uniformly distributed point sets

Area of Science:

• Applied Mathematics
• Computational Science

Background:

• Low-discrepancy point sets are crucial for experimental design, numerical integration, computer graphics, and finance.
• Recent advancements utilized Graph Neural Networks and solver-based optimization for improved point set construction.

Purpose of the Study:

• To develop novel methods for constructing low-discrepancy point sets with substantially lower discrepancy.
• To improve upon existing constructions, including those by Rusch et al. (2024).

Main Methods:

• Separating point set construction into relative point positioning and optimal placement.
• Utilizing tailored permutations to optimize point relationships and placement.
• Evaluating discrepancy reduction compared to previous methods.

Main Results:

• Achieved point sets with 20% lower average discrepancy compared to Rusch et al.
• Reduced the number of points required in 2D to achieve a discrepancy of 0.005 from over 500 to under 350.
• Demonstrated significant efficiency gains for querying time-consuming models.

Conclusions:

• The proposed method offers substantial improvements in low-discrepancy point set construction.
• This advancement leads to significant reductions in computational cost for various applications.
• Further optimization of point set generation is achievable through strategic construction approaches.