Holomorphic Cauchy Networks for Efficient Stationary Tail Computation in High-Dimensional Reflected Brownian Motions
Abstract
Stationary tail computation for high-dimensional reflected Brownian motions (RBMs) remains computationally challenging. Building on NN4MGF's shared-coordinate basic-adjoint-relationship (BAR) framework, we develop Holomorphic Cauchy Networks (HCNs). Native complex-valued rational Cauchy subnetworks enforce holomorphicity on pole-free domains, removing sampled Cauchy-Riemann penalties and their input-derivative graphs. Exact origin anchoring and boundary-coordinate exclusion enforce structural identities, while anchored interactions represent dependent transforms. Under holomorphic-log assumptions, we establish simultaneous value and first-derivative approximation for the constrained architectures and a mixed-derivative lower bound for coordinate-only representations under nonzero cross-covariance. Closed-form real-axis derivatives and sequential backpropagation further control memory. On the same NVIDIA H100 NVL, the tested 30D HCN configuration uses approximately tenfold less peak allocated memory and has 9.7-fold lower mean update time than NN4MGF; extrapolation to the respective scheduled update counts gives a projected full-schedule update-time ratio of 58.2. With 64 million sampled base inputs per high-dimensional run, HCN reaches 50 dimensions on a laptop GPU with 6.367% mean relative error over thresholds 1 through 8. Degree-matched comparisons yield lower reported mean relative errors over the primary evaluation ranges, including 1.422% versus 6.684% on the 2D Harrison non-product benchmark. An exploratory 10D correlated example illustrates the interaction extension.
Related
- Entropic metastability for drifted reflected Brownian motion: quasi-stationary distributions, regularity, and the exit position distribution
- Quantitative stratification and optimal regularity for stationary J-holomorphic maps
- Limit theorems for one-dimensional diffusions: reflected, skew, sticky, skew-sticky, and snapping-out limits
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BibTeX
@misc{wang2026holomorphic,
title = {{Holomorphic Cauchy Networks for Efficient Stationary Tail Computation in High-Dimensional Reflected Brownian Motions}},
author = {Kehan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36558},
eprint = {2610.36558},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36558},
}
RIS
TY - UNPB
TI - Holomorphic Cauchy Networks for Efficient Stationary Tail Computation in High-Dimensional Reflected Brownian Motions
AU - Kehan Wang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36558
UR - https://arxiv.org/abs/2610.36558
AB - Stationary tail computation for high-dimensional reflected Brownian motions (RBMs) remains computationally challenging. Building on NN4MGF's shared-coordinate basic-adjoint-relationship (BAR) framework, we develop Holomorphic Cauchy Networks (HCNs). Native complex-valued rational Cauchy subnetworks enforce holomorphicity on pole-free domains, removing sampled Cauchy-Riemann penalties and their input-derivative graphs. Exact origin anchoring and boundary-coordinate exclusion enforce structural identities, while anchored interactions represent dependent transforms. Under holomorphic-log assumptions, we establish simultaneous value and first-derivative approximation for the constrained architectures and a mixed-derivative lower bound for coordinate-only representations under nonzero cross-covariance. Closed-form real-axis derivatives and sequential backpropagation further control memory. On the same NVIDIA H100 NVL, the tested 30D HCN configuration uses approximately tenfold less peak allocated memory and has 9.7-fold lower mean update time than NN4MGF; extrapolation to the respective scheduled update counts gives a projected full-schedule update-time ratio of 58.2. With 64 million sampled base inputs per high-dimensional run, HCN reaches 50 dimensions on a laptop GPU with 6.367% mean relative error over thresholds 1 through 8. Degree-matched comparisons yield lower reported mean relative errors over the primary evaluation ranges, including 1.422% versus 6.684% on the 2D Harrison non-product benchmark. An exploratory 10D correlated example illustrates the interaction extension.
ER -