Improve the topology and sample-complexity analysis for random Lagrangian sensors

Develop a better topology for coefficient functions that exploits Monte Carlo convergence rates and thereby improves the curse-of-dimensionality estimate for the number of random Lagrangian sensors required to resolve the coefficient family.

Background

The paper proves that random Lagrangian probe sensors resolve a compact coefficient family with high probability, but the proof uses an h-net argument. The resulting estimate for the number of sensors has curse-of-dimensionality behavior and does not exploit the Monte Carlo nature of the random probes.

The authors explicitly identify the need for a topology on coefficient functions that can make use of Monte Carlo rates. Resolving this issue would sharpen the theoretical sensor-resolution requirements for the Lagrangian attention sampler.

References

We remark that the estimate of $m$ in the proof of Theorem \ref{thm:randomFS} in Appendix \ref{app:express} has curse of dimensionality. The proof in Appendix \ref{app:express} is based on the $h$-net and does not take advantage of the Monte Carlo feature of the method, so it should have overestimated $m$. The improvement needs a better topology for the coefficient functions that could make use of the Monte Carlo rate. This is left for future study.

— Deep operator learning for efficient sampling from invariant measures of stochastic differential equations  (2609.11376 - Guo et al., 10 Sep 2026) in Remark following Theorem 3.8 (Theorem \ref{thm:randomFS}), Section 4.1, and Appendix \ref{app:express}