A search for short-period Tausworthe generators over $\mathbb{F}_b$ with application to Markov chain quasi-Monte Carlo

TL;DR

The paper investigates practical driving sequences for Markov chain QMC by extending Harase's $\mathbb{F}_2$ search to finite fields $\mathbb{F}_b$ and developing a Fibonacci-polynomial-based search to find maximal-period Tausworthe generators with $t$-value zero for $s=3$ and small values for higher dimensions. It provides an explicit parameter catalog for $\mathbb{F}_4$ and compares these new generators to existing $\mathbb{F}_2$-based approaches, showing competitive or improved performance in Gibbs sampling, queuing, and Bayesian linear regression tasks. The results support using the $t$-value criterion as a robust guide for selecting driving sequences in Markov chain QMC and motivate exploration of additional $\mathbb{F}_b$-based QMC constructions. The work broadens the practical toolkit for QMC-enabled Bayesian computation and highlights the value of continued fractions and polynomial-analytic methods in generator design.

Abstract

A one-dimensional sequence $u_0, u_1, u_2, \ldots \in [0, 1)$ is said to be completely uniformly distributed (CUD) if overlapping $s$-blocks $(u_i, u_{i+1}, \ldots , u_{i+s-1})$, $i = 0, 1, 2, \ldots$, are uniformly distributed for every dimension $s \geq 1$. This concept naturally arises in Markov chain quasi-Monte Carlo (QMC). However, the definition of CUD sequences is not constructive, and thus there remains the problem of how to implement the Markov chain QMC algorithm in practice. Harase (2021) focused on the $t$-value, which is a measure of uniformity widely used in the study of QMC, and implemented short-period Tausworthe generators (i.e., linear feedback shift register generators) over the two-element field $\mathbb{F}_2$ that approximate CUD sequences by running for the entire period. In this paper, we generalize a search algorithm over $\mathbb{F}_2$ to that over arbitrary finite fields $\mathbb{F}_b$ with $b$ elements and conduct a search for Tausworthe generators over $\mathbb{F}_b$ with $t$-values zero (i.e., optimal) for dimension $s = 3$ and small for $s \geq 4$, especially in the case where $b = 3, 4$, and $5$. We provide a parameter table of Tausworthe generators over $\mathbb{F}_4$, and report a comparison between our new generators over $\mathbb{F}_4$ and existing generators over $\mathbb{F}_2$ in numerical examples using Markov chain QMC.

Figures (8)

• Figure 1: Distribution of orthogonal multiplicities $M(p)$ for all monic irreducible polynomials $p(x) \in \mathbb{F}_b[x]$ with $\deg(p\textcolor{red}{(x)})=m$.
• Figure 2: Initial part of the tree of Fibonacci polynomials over $\mathbb{F}_3$.
• Figure 3: RMSEs for $E[X_1]$, $E[X_2]$, and $E[X_3]$ with true value $0$.
• Figure 4: RMSEs for $E[X_1X_2]$, $E[X_1X_3]$, and $E[X_2X_3]$ with true values 0.3, -0.2, and 0.5.
• Figure 5: RMSEs for $E[X_1 X_2 X_3]$ with true value $0$.

Theorems & Definitions (15)

• Definition 2.1: Discrepancy
• Definition 2.2: CUD sequences
• Theorem 2.3: CHENTSOV1967218
• Definition 2.4: Tausworthe generators over $\mathbb{F}_b$
• Definition 2.5: $(t, m, s)$-nets
• Theorem 2.6: MR1172997MR1160278
• Remark 1
• Theorem 3.1
• Remark 2
• Remark 3