Scrambled Halton Subsequences and Inverse Star-Discrepancy (2411.10363v2)

Abstract: Braaten and Weller discovered that the star-discrepancy of Halton sequences can be strongly reduced by scrambling them. In this paper, we apply a similar approach to those subsequences of Halton sequences which can be identified to have low-discrepancy by results from $p$-adic discrepancy theory. For given finite $N$, it turns out that the star-discrepancy of these sequences is surprisingly low. By that known empiric bounds for the inverse star-discrepancy can be improved. Furthermore, we establish the existence of $N$-point sets in dimension $d$ whose star-discrepancy satisfies $\leq 2.4631832 \sqrt{\frac{d}{N}}$, where the constant improves upon all previously known bounds.

• The paper demonstrates that scrambling Halton subsequences reduces star-discrepancy via p-adic methods.
• It establishes rigorous discrepancy bounds in both single and multidimensional settings that outperform current techniques.
• The research offers practical frameworks for improving high-dimensional integration and advances discrepancy theory.

Analysis of Christian Weiß's "Scrambled Halton Subsequences and Inverse Star-Discrepancy"

Christian Weiß's paper addresses an important question in discrepancy theory, specifically focusing on the star-discrepancy of Halton sequences and their subsequences. The work builds on the foundational idea established by Braaten and Weller that scrambling Halton sequences can significantly decrease their star-discrepancy. Weiß extends this notion to the subsequences of the Halton sequences, revealing potentially lower discrepancies than those previously recorded.

Core Contributions

The paper primarily contributes to the understanding of the relationship between low-discrepancy sequences and their subsequences, positing that subsequences derived from a Halton sequence retain or improve the discrepancy properties of the original sequence. This exploration into discrepancy theory is done through the lens of p-adic theory, tying back to fundamental concepts in number theory and the asymptotic properties of uniformly distributed sequences.

Key Theorems and Results:

1. Theorem 1: This theorem establishes a bound on the discrepancy of scrambled van der Corput subsequences, which are a specific type of low-discrepancy sequence generated through permutation polynomials modulo a prime number.
2. Theorem 2: It extends the application of scrambled sequences to multi-dimensional contexts. This result is particularly significant as it proposes that through careful choice of permutations and shifts, the subsequences of Halton sequences can achieve particularly low star-discrepancies.
3. Theorem 4: This theorem is refined for small dimensions using Halton sequences and improved bounds for discrepancy to illustrate that certain point sets have unexpectedly low discrepancies.

These results indicate that the empirical and theoretical exploration into scrambled Halton subsequences offers practical methodologies for producing sequences with remarkably low discrepancies. The numerical comparisons provided in the paper demonstrate that scrambled Halton subsequences outperform or equal current state-of-the-art techniques across small to moderate dimensions.

Practical and Theoretical Implications

This research carries implications for high-dimensional integration tasks, particularly those utilizing quasi-Monte Carlo methods, where minimizing discrepancy directly corresponds to minimizing integration error according to the Koksma-Hlawka inequality. Practically, this implies more efficient computation of integrals using fewer sample points, significantly impacting fields such as computational finance and physics, where Monte Carlo simulations are heavily relied upon.

Theoretically, this paper challenges current understandings of the best-possible discrepancy bounds for low-discrepancy sequences, potentially opening avenues for further research into p-adic discrepancy theory and the development of new sequences with even lower discrepancies. It also emphasizes scrutiny on the role of prime number bases and permutations in achieving low-discrepancy sequences, facilitating a deeper understanding of their optimal configurations.

Future Directions

This research sets the stage for several potential future developments. One direction could involve refining the scrambling techniques for multidimensional sequences, particularly in dimensions exceeding those explored in Weiß's paper. Additionally, investigating relationships with secure pseudorandom bit generators could enrich both theoretical insights and application scopes. Researchers may also extend the computational frameworks used in this paper to achieve more scalable solutions as computational resources improve.

Overall, Weiß's paper deepens the understanding of scrambling techniques in the field of low-discrepancy sequence construction. By advancing both empirical outcomes and theoretical frameworks, this work forms a substantial contribution to discrepancy theory and its associated computational methodologies.