Improved Error Rates for QMC Slicing

Definitions and mathematical statement

Let $m\in{\mathbb{N}}$ denote the number of directions used in a sliced approximation. For the sole purpose of describing the requested comparison, write $S$ for the kernel sum being approximated, $S_m^{\mathrm{QMC}}$ for its sliced approximation using a quasi-Monte Carlo design, and $S_m^{\mathrm{ind}}$ for the corresponding approximation using independent random directions. These are schematic labels: the supplied record does not define the kernel sum, the slicing construction, the direction space, the quasi-Monte Carlo designs, or the distributions of the independent directions.

Requested problem. Establish rigorous convergence-rate estimates, as $m\to\infty$, explaining the improved errors observed numerically for $S_m^{\mathrm{QMC}}$ compared with $S_m^{\mathrm{ind}}$.

Unresolved formulation. The record supplies neither a numerical rate nor an error criterion. In particular, it does not specify whether the comparison concerns deterministic error, expected error, mean-square error, a probability bound, or another quantity. It also omits the admissible kernels, data configurations, dimensions, regularity assumptions, and permitted dependence of constants. Consequently, this request cannot yet be expressed as a complete quantified theorem without additional source information. The notation above introduces no assumptions filling these gaps.

The supplied registry labels the problem open and reports that the primary account leaves the proof open, while later work presents the improvement as numerical evidence. These are supplied curation claims, not independently verified mathematical or literature conclusions.

Short English statement

The requested result is a proof of the numerically observed convergence improvement obtained by replacing independent random directions with quasi-Monte Carlo designs in sliced kernel-sum approximations. The supplied record leaves the construction, error measure, hypotheses, and target rates unspecified, so a precise theorem remains undetermined.

Sources

• Local UnsolvedMath registry: supplied record for permanent ID $30006083$, source code catalogue entry, in the collection associated with data/unsolvedmath/problems.json and catalog/problems.json. No individual proposer is supplied; the proposed year is recorded as 2024.
• Supplied source citation: Deep Learning for PDE-based Inverse Problems (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/48, pp. 2805--2900. https://doi.org/10.4171/owr/2024/48. The background attributes the extraction to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source: Oberwolfach Report 48/2024. https://doi.org/10.4171/owr/2024/48.
• Supplied literature source: Hertrich--Jahn--Quellmalz, Fast summation by QMC slicing. https://openreview.net/pdf?id=iNmVX9lx9l.
• Literature assessment dated exactly 2026-08-22: the supplied curation describes the requested proof as unspecified, reports that the primary account explicitly leaves it open, and describes later work as providing numerical evidence. This is supplied dated curation, not independently verified; its time-sensitive status and cited claims require independent checking.
• The registry assigns the statement status corrected_verified and states that it was reconstructed and checked against the official report. This is a supplied verification assertion; no source was independently consulted for this draft.