Recent developments on Array-RQMC

We use Array-RQMC sampling in a walk on spheres (WoS) algorithm for Dirichlet boundary value problems. On a collection of problems, we find that Array-RQMC-WoS reduces the Monte Carlo MSE or variance by factors ranging from $71$ -fold to $3087$ -fold at $n=2^{17}$ trajectories. The variance is known to be $o(1/n)$ but attains empirical rates between $n^{-1.4}$ and $n^{-1.8}$ in our examples. A simpler RQMC-WoS algorithm studied in Ho and Owen (2026) has more theoretical support but only reduced variance by 1.8 to 10.7-fold on the same set of examples. In order to explain this improvement, we introduce a column-wise mean dimension of the RQMC error based on Sobol' indices. It matches the usual mean dimension for Monte Carlo and the mean dimension of a dual lattice error for randomized lattices. We find for a gasket example from Crane et al. (2025) that the mean dimension of Array-RQMC-WoS errors is much higher than an analogous Array-MC-WoS algorithm has. v2 replaced v1's QMCPy lattice with Korobov lattices from LatNet Builder, but left the old abstract in the meta-data v3 corrects the v2 abstract in this meta dataWe use Array-RQMC sampling in a walk on spheres (WoS) algorithm for Dirichlet boundary value problems. On a collection of problems, we find that Array-RQMC-WoS reduces the Monte Carlo MSE or variance by factors ranging from $71$-fold to $3087$-fold at $n=2^{17}$ trajectories. The variance is known to be $o(1/n)$ but attains empirical rates between $n^{-1.4}$ and $n^{-1.8}$ in our examples. A simpler RQMC-WoS algorithm studied in Ho and Owen (2026) has more theoretical support but only reduced variance by 1.8 to 10.7-fold on the same set of examples. In order to explain this improvement, we introduce a column-wise mean dimension of the RQMC error based on Sobol' indices. It matches the usual mean dimension for Monte Carlo and the mean dimension of a dual lattice error for randomized lattices. We find for a gasket example from Crane et al. (2025) that the mean dimension of Array-RQMC-WoS errors is much higher than an analogous Array-MC-WoS algorithm has. v2 replaced v1's QMCPy lattice with Korobov lattices from LatNet Builder, but left the old abstract in the meta-data v3 corrects the v2 abstract in this meta data

arXiv:2605.12844May 13, 20260 citations

Fixed-Horizon Error Rates for Array-RQMC Metropolis-Hastings

Jiarui Du, Josef Dick, Zexin Pan +1 more

This paper studies randomised quasi-Monte Carlo (RQMC) methods for Metropolis--Hastings (MH) algorithms, whose accept--reject discontinuity complicates RQMC analysis. To circumvent some of these complications, we study Array-RQMC for MH, with $N$ replicated chains evolved over a fixed time horizon $T$ , where we make use of Hilbert sorting for joint multidimensional updates at each step. For one-dimensional MH and bounded-variation test functions, we prove a sampling mean squared error bound of $\mathcal O_T(N^{-4/3})$ relative to the finite-horizon expectation. For product targets, coordinatewise Array-RQMC Metropolis-within-Gibbs retains this exponent for additive marginal bounded-variation tests, with constants that may depend on dimension. With Gaussian random-walk proposals, the bounds extend to scalar and additive marginal polynomials with an additional factor $(\log N)^\ell$ , where $\ell$ is the maximum polynomial degree. For joint updates in state dimension $p$ using $p$ proposal uniforms, we provide a conditional $\mathcal O_T(N^{-1-1/(p+2)})$ framework under regularity assumptions that are not verified for the joint multidimensional examples. Numerical experiments support the verified one-dimensional theory and demonstrate variance reduction for joint Hilbert-sorted updates, including Bayesian logistic regression, while illustrating the limitations of these gains as dimension increases.This paper studies randomised quasi-Monte Carlo (RQMC) methods for Metropolis--Hastings (MH) algorithms, whose accept--reject discontinuity complicates RQMC analysis. To circumvent some of these complications, we study Array-RQMC for MH, with $N$ replicated chains evolved over a fixed time horizon $T$, where we make use of Hilbert sorting for joint multidimensional updates at each step. For one-dimensional MH and bounded-variation test functions, we prove a sampling mean squared error bound of $\mathcal O_T(N^{-4/3})$ relative to the finite-horizon expectation. For product targets, coordinatewise Array-RQMC Metropolis-within-Gibbs retains this exponent for additive marginal bounded-variation tests, with constants that may depend on dimension. With Gaussian random-walk proposals, the bounds extend to scalar and additive marginal polynomials with an additional factor $(\log N)^\ell$, where $\ell$ is the maximum polynomial degree. For joint updates in state dimension $p$ using $p$ proposal uniforms, we provide a conditional $\mathcal O_T(N^{-1-1/(p+2)})$ framework under regularity assumptions that are not verified for the joint multidimensional examples. Numerical experiments support the verified one-dimensional theory and demonstrate variance reduction for joint Hilbert-sorted updates, including Bayesian logistic regression, while illustrating the limitations of these gains as dimension increases.

arXiv:2610.36431Oct 27, 20260 citations

Prediction and Classification with minimal QRC and QELMs

Christoph Räth

Jan 1, 20260 citations

The Cluster Expansion

Lothar Schäfer

Springer eBooksJan 1, 19990 citations

Exact State-Ranked Array-RQMC: Representations and Shared Work

Aoi Kawasaki

Repeated states in state-ranked Array-RQMC receive rank-specific inputs. We execute the reranked population without visiting every particle when complete states repeat and transitions admit short finite-word interval partitions. Primal image-basis and dual constraint counters preserve the realized histogram path and supported readouts. Fixed suffix dependencies survive triangular scrambles; syndrome-prefix diversity determines exact conditional demand for shared dual transformations. A sharp finite-sum bound and rank-boundary geometry connect demand to execution cost. Six compiled executors share repair and tandem tapes and return every step histogram and readout. With eight seeds per cell on one non-isolated host, median within-seed timing ratios favor direct basis over the other non-enumerating methods in all 17 cells. With 2^20 particles, it outpaces compact rank streaming by about 41-57 times for repair and 10 for tandem. Dual coefficient retention improves on matched reconstruction; fixed-workload diagnostics characterize the remaining work. This record contains the 20-page author manuscript and 6-page reader supplement, version 20261005-estimator. This is an author preprint; it has not yet been peer reviewed or accepted for publication. Code and reproduction data. The fixed public companion exact-state-ranked-array-rqmc-v1.0.0 provides the measured source, all 4,080 original timings, saved trajectories, corrected executors, and table, figure and PDF reproduction. Download array-rqmc-repro-20261006.zip and follow its README. It is the public counterpart of array-rqmc-review-20261005-estimator.zip named in the paper; its provenance file identifies the unchanged scientific inputs and the omission of private editorial material. Code and the layout template use MPL-2.0; author publication materials and saved data use CC BY 4.0. Dependencies retain their own licenses. The two PDFs deposited here remain unchanged.Repeated states in state-ranked Array-RQMC receive rank-specific inputs. We execute the reranked population without visiting every particle when complete states repeat and transitions admit short finite-word interval partitions. Primal image-basis and dual constraint counters preserve the realized histogram path and supported readouts. Fixed suffix dependencies survive triangular scrambles; syndrome-prefix diversity determines exact conditional demand for shared dual transformations. A sharp finite-sum bound and rank-boundary geometry connect demand to execution cost. Six compiled executors share repair and tandem tapes and return every step histogram and readout. With eight seeds per cell on one non-isolated host, median within-seed timing ratios favor direct basis over the other non-enumerating methods in all 17 cells. With 2^20 particles, it outpaces compact rank streaming by about 41-57 times for repair and 10 for tandem. Dual coefficient retention improves on matched reconstruction; fixed-workload diagnostics characterize the remaining work. This record contains the 20-page author manuscript and 6-page reader supplement, version 20261005-estimator. This is an author preprint; it has not yet been peer reviewed or accepted for publication. Code and reproduction data. The fixed public companion exact-state-ranked-array-rqmc-v1.0.0 provides the measured source, all 4,080 original timings, saved trajectories, corrected executors, and table, figure and PDF reproduction. Download array-rqmc-repro-20261006.zip and follow its README. It is the public counterpart of array-rqmc-review-20261005-estimator.zip named in the paper; its provenance file identifies the unchanged scientific inputs and the omission of private editorial material. Code and the layout template use MPL-2.0; author publication materials and saved data use CC BY 4.0. Dependencies retain their own licenses. The two PDFs deposited here remain unchanged.

Oct 6, 20260 citations

71

-fold to

3087

-fold at

n=2^{17}

trajectories. The variance is known to be

o(1/n)

but attains empirical rates between

n^{-1.4}

and

n^{-1.8}

in our examples. A simpler RQMC-WoS algorithm studied in Ho and Owen (2026) has more theoretical support but only reduced variance by 1.8 to 10.7-fold on the same set of examples. In order to explain this improvement, we introduce a column-wise mean dimension of the RQMC error based on Sobol' indices. It matches the usual mean dimension for Monte Carlo and the mean dimension of a dual lattice error for randomized lattices. We find for a gasket example from Crane et al. (2025) that the mean dimension of Array-RQMC-WoS errors is much higher than an analogous Array-MC-WoS algorithm has. v2 replaced v1's QMCPy lattice with Korobov lattices from LatNet Builder, but left the old abstract in the meta-data v3 corrects the v2 abstract in this meta data

N

replicated chains evolved over a fixed time horizon

T

, where we make use of Hilbert sorting for joint multidimensional updates at each step. For one-dimensional MH and bounded-variation test functions, we prove a sampling mean squared error bound of

\mathcal O_T(N^{-4/3})

relative to the finite-horizon expectation. For product targets, coordinatewise Array-RQMC Metropolis-within-Gibbs retains this exponent for additive marginal bounded-variation tests, with constants that may depend on dimension. With Gaussian random-walk proposals, the bounds extend to scalar and additive marginal polynomials with an additional factor

(\log N)^\ell

, where

\ell

is the maximum polynomial degree. For joint updates in state dimension

p

using

p

proposal uniforms, we provide a conditional

\mathcal O_T(N^{-1-1/(p+2)})

framework under regularity assumptions that are not verified for the joint multidimensional examples. Numerical experiments support the verified one-dimensional theory and demonstrate variance reduction for joint Hilbert-sorted updates, including Bayesian logistic regression, while illustrating the limitations of these gains as dimension increases.