# Holomorphic Cauchy Networks for Efficient Stationary Tail Computation in High-Dimensional Reflected Brownian Motions

DevFeed: [Holomorphic Cauchy Networks for Efficient Stationary Tail Computation in High-Dimensional Reflected Brownian Motions](<https://devfeed.tech/articles/holomorphic-cauchy-networks-for-efficient-stationary-tail-computation-in-high-dimensional-reflected-brownian-motions-103970.md>)

Original publisher: [Read original article](<https://arxiv.org/abs/2610.36558>)

Author: Kehan Wang

Published: 2026-10-28T04:00:00Z

Content type: article

Language: en

Sources: [arXiv: Machine Learning](<https://devfeed.tech/sources/arxiv-machine-learning.md>)

Topics: [Embeddings](<https://devfeed.tech/topics/embeddings.md>), [GPU](<https://devfeed.tech/topics/gpu.md>), [Supercomputing](<https://devfeed.tech/topics/supercomputing.md>)

Tags: [approximation](<https://devfeed.tech/tags/approximation.md>), [cs-lg](<https://devfeed.tech/tags/cs-lg.md>), [cs-na](<https://devfeed.tech/tags/cs-na.md>), [machine-learning](<https://devfeed.tech/tags/machine-learning.md>), [math-na](<https://devfeed.tech/tags/math-na.md>), [math-pr](<https://devfeed.tech/tags/math-pr.md>), [networks](<https://devfeed.tech/tags/networks.md>)

## AI overview

Holomorphic Cauchy Networks (HCNs) target stationary tail computation for high-dimensional reflected Brownian motions by extending NN4MGF's shared-coordinate BAR framework. Their complex rational subnetworks enforce holomorphicity on pole-free domains, avoiding sampled Cauchy-Riemann penalties; origin anchoring, boundary-coordinate exclusion, and anchored interactions encode structural identities and dependencies. On the same NVIDIA H100 NVL, the tested 30D HCN setup used about one tenth the peak allocated memory and had 9.7 times lower mean update time than NN4MGF; the 58.2-fold full-schedule update-time ratio is an extrapolation. With 64 million sampled base inputs per high-dimensional run, HCN reached 50 dimensions on a laptop GPU at 6.367% mean relative error over thresholds 1-8. Degree-matched comparisons reported lower mean relative errors over primary evaluation ranges, including 1.422% versus 6.684% on the 2D Harrison non-product benchmark. The abstract also reports theoretical approximation results under holomorphic-log assumptions and an exploratory 10D correlated example.

## Source excerpt

arXiv:2610.36558v1 Announce Type: new 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.