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

Kehan Wang

arXiv:2610.36558v1cs.LGcs.NAmath.NAmath.PR

TL;DR

High-dimensional RBM stationary-tail computation remains challenging, and BAR-based transform learning faces memory and runtime barriers. This paper develops HCN, which builds holomorphicity and structural constraints into shared-coordinate networks. Experiments report lower update time and memory use, alongside lower errors in degree-matched comparisons.

  • Problem

    Accurate stationary-tail computation for high-dimensional RBMs remains computationally challenging, while NN4MGF’s GPU memory demand and training time impede scaling.

    Source: §1 · p. 2

  • Method

    HCN embeds native complex rational Cauchy subnetworks and exact structural constraints in NN4MGF’s shared-coordinate BAR framework, adding anchored interactions for dependent transforms.

    Source: §1 · p. 2, §6 · p. 17, §7 · p. 18

  • Results

    HCN reports lower mean relative errors than NN4MGF over degree-matched primary evaluation ranges, including 1.422% versus 6.684% on the 2D Harrison non-product benchmark, and 9.7-fold lower mean update time in the 30D H100 profile.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The reported experiments demonstrate that HCN can scale to 50 dimensions on a laptop GPU using 64 million sampled base inputs per high-dimensional run, with 6.367% mean relative error over thresholds 1 through 8.

    Source: §1 · p. 3

    The approximation theorem establishes existence at finite but unrestricted width, but does not guarantee optimization convergence at fixed trained width or certify that learned transforms are globally valid probability measures.

    Source: §6 · p. 18

Abstract

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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.

1 Introduction

High-dimensional RBM stationary-tail computation remains costly, and NN4MGF’s BAR-based transform learning faces memory and runtime barriers as dimension grows. HCN embeds holomorphicity and structural identities in the architecture, with approximation guarantees and experiments reporting improved efficiency and tail accuracy.

HCN architecture

  • HCN uses native complex-valued rational Cauchy subnetworks to enforce holomorphicity on pole-free domains, removing sampled Cauchy–Riemann penalties and their input-derivative graphs.This retains NN4MGF’s shared-coordinate BAR framework while replacing its separate real and imaginary output heads and Fourier feature encodings.Source: §1 · p. 2

Structural and computational refinements

  • Exact origin anchoring and boundary-coordinate exclusion enforce transform identities algebraically, while anchored interaction subnetworks represent dependent transforms.Closed-form real-axis derivatives and sequential backpropagation further control memory during training.Source: §1 · p. 2

Approximation properties

  • Under holomorphic-log assumptions, HCN approximates target values and first derivatives, while nonzero cross-covariance creates an irreducible uniform error for coordinate-only representations.The lower bound cannot be removed by increasing width alone.Source: §1 · pp. 2–3

Numerical experiments

  • On the 30D NVIDIA H100 NVL profile, HCN uses approximately tenfold less peak allocated memory and has 9.7-fold lower mean update time than NN4MGF.Extrapolating measured times to scheduled update counts gives a projected full-schedule update-time ratio of 58.2; degree-matched comparisons also report lower mean relative errors, including 1.422% versus 6.684% on the 2D Harrison non-product benchmark.Source: §1 · p. 3
  • Using 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.The experiments also include an exploratory 10D correlated example illustrating the interaction extension.Source: §1 · p. 3

Paper organization

  • The paper is organized into sections on the RBM transform problem, HCN design and approximation, memory-controlled training, experiments, and scope and conclusions.Proofs, implementation details, and additional diagnostics are in the appendices.Source: §1 · p. 3

2 RBM Stationary Tails and Transform Learning

The paper formulates stationary workload-tail computation through RBM transforms and the basic adjoint relationship (BAR). It learns those transforms from system parameters and applies numerical Laplace inversion to recover tails.

2.1 The stationary tail problem

  • The target is the stationary tail probability of total workload, computed directly from system primitives without generating stationary sample paths.The RBM is assumed to have a unique stationary distribution and finite first moments under the stated model conditions.Source: §2.1 · p. 3

2.2 Transforms and the basic adjoint relationship

  • The interior and boundary Laplace transforms are holomorphic in their common convergence domain, and the BAR couples them through the model parameters (µ,Σ,R).Boundary transforms are structurally independent of the coordinate normal to their supporting face.Source: §2.2 · p. 4

2.3 From learned transforms to tails

  • The learned interior transform restricted to the complex diagonal gives the total-workload transform, whose tail transform is recovered by numerical Laplace inversion.The method uses Fixed-Talbot inversion, which evaluates the transform at complex quadrature nodes and therefore requires analyticity and numerical stability across complex arguments.Source: §2.3 · p. 4

3 Holomorphic Cauchy Networks

HCN parameterizes RBM log transforms with native complex Cauchy subnetworks and exact structural constraints. Shared coordinate blocks capture separable targets, anchored interactions extend them to dependent targets, and the paper establishes constrained approximation results while specifying training losses.

3 Holomorphic Cauchy Networks

  • Native complex Cauchy subnetworks, origin anchors, and boundary-coordinate exclusion define log transforms that are holomorphic on pole-free domains and satisfy prescribed transform values and boundary independence exactly.The BAR and real-axis constraints are imposed through the training loss.Source: §3 · p. 5, §3.1 · p. 5

3.1 Complex Cauchy representation

  • The rational Cauchy activation is extended to complex inputs with trainable complex coefficients and scale parameters that regulate pole locations.The resulting networks and exponentiated transforms are holomorphic wherever their Cauchy denominators are nonzero.Source: §3.1 · p. 5

3.2 Shared coordinate representations and interactions

  • Shared coordinate subnetworks provide separable interior and boundary representations, with parameter count linear in dimension at fixed widths; anchored interaction subnetworks extend the architecture to dependent transforms.Mixing coordinates before the Cauchy activation allows cross-coordinate interactions while retaining the exact origin and boundary-coordinate identities.Source: §3.2 · p. 6

3.3 Approximation with shared blocks and exact constraints

  • Under holomorphicity, origin-value, and boundary-independence conditions, finite-width blocks simultaneously approximate values and first derivatives for both separable and general dependent targets.The construction preserves exact identities and has a uniform denominator margin; nonzero cross-covariance yields a uniform approximation obstruction for coordinate-only representations.Source: §3.3 · pp. 6–7

3.4 Training constraints

  • The training constraints combine BAR residuals, pairwise boundary residuals, and real-axis monotonicity constraints, with equal weights in reported experiments.Boundary residuals are evaluated directly in transform space, avoiding complex logarithms of BAR coefficients.Source: §3.4 · p. 8

4 Training under GPU Memory Constraints

HCN removes the sampled Cauchy–Riemann penalty, but BAR evaluation still couples multiple transforms and the real-axis monotonicity loss still requires input derivatives. The training procedure limits memory through explicit derivatives and sequential loss-branch backpropagation.

4 Training under GPU Memory Constraints

  • HCN eliminates the Cauchy–Riemann penalty and its derivative graph, while BAR residuals still couple d +1 transforms and the monotonicity penalty requires first-order coordinate derivatives.The section addresses these remaining costs with closed-form derivatives and sequential backpropagation.Source: §4 · p. 8

4.1 Explicit input derivatives

  • Input derivatives are computed explicitly only for the real-axis monotonicity loss; BAR and isolated boundary losses need forward evaluations, while parameter gradients still use reverse-mode autodiff.The explicit Jacobian replaces the inner automatic-differentiation pass used to construct input derivatives.Source: §4.1 · p. 9

4.2 Sequential backpropagation and the sampling budget

  • The 30D and 50D implementations backpropagate the BAR, boundary, and combined real-axis branches in sequence, releasing each graph before constructing the next.For fixed parameters, samples, and detached BAR weights, accumulated gradients match a combined backward pass up to floating-point arithmetic while reducing simultaneous storage of branch-specific intermediates.Source: §4.2 · p. 9
  • The 10D implementation accumulates boundary faces separately, whereas 2D and 20D use a combined backward call; these are graph-scheduling choices, not changes to the training objective.In 10D, the boundary-face accumulation retains the 1/d factor.Source: §4.2 · p. 9
  • Dimension increases reduce per-update batch sizes and increase optimizer updates to maintain a 64 million sampled-base-input budget per high-dimensional run.The count covers freshly sampled transform-argument vectors; derived arguments and origin anchors are excluded.Source: §4.2 · p. 9

4.3 Sampling and optimization

  • BAR batches mix general complex, complex-diagonal, near-diagonal, and nonnegative real-diagonal inputs, while boundary batches use mapped complex samples and real-axis batches use real-part projections.HCN runs use complex128 and Adam with a learning rate of 5×10−5 for the first 80% of updates and 10−5 thereafter, retaining optimizer state across the transition.Source: §4.3 · p. 10

5 Numerical Experiments

Across product-form and non-product benchmarks from 2D to 50D, HCN reports lower errors in degree-matched primary comparisons and reduced resource use in the tested 30D profile. The experiments also examine inversion degree, activation choice, late-training stability, and an exploratory 10D correlated model.

5.1 Benchmarks and evaluation protocol

  • Experiments cover 2D–50D benchmarks, use fixed training endpoints, and evaluate networks trained solely from RBM parameters, with references reserved for post-training evaluation.The 20D results average three seeds; inversion degree is varied without retraining the network.Source: §5 · p. 10, §5.1 · p. 11

5.2 Computational performance and dimensional scaling

  • At 30D on the same NVIDIA H100 NVL, HCN uses 7.805 GiB versus 77.815 GiB peak allocated memory and averages 0.165113 s versus 1.601046 s per update.The projected full-schedule update-time ratio is 58.2, based on respective scheduled update counts rather than completed runs or time to a common accuracy target.Source: Table 2 · p. 11, §5.2 · pp. 11–12

5.3 Tail accuracy and the 50D extension

  • Across degree-matched primary ranges, HCN reports lower MRE than NN4MGF, including 1.422% versus 6.684% on the 2D Harrison benchmark.Table 3 compares tail errors at fixed endpoints.Source: Table 3 · p. 12, §5.3 · pp. 12–13
  • The 50D HCN run completed on a single laptop GPU and achieved MRE1:8 of 6.367%.The run used 64 million sampled base inputs; its MRE1:5 and MRE1:10 were 1.495% and 8.011%.Source: §5.2 · p. 12, §5.3 · p. 13

5.4 Inversion and activation diagnostics

  • Degree-5 Fixed-Talbot inversion has substantial high-dimensional error, including 23.898% at 50D on 1:8; Degree 8 reduces it to 0.414%.At 20D, Degree-5 quadrature bias partially cancels NN4MGF transform error; Degree 8 is adopted as the primary high-dimensional metric.Source: Table 4 · p. 14, §5.4 · p. 14
  • With the same 20D coordinate architecture, seeds, input budget, and endpoint, Cauchy activation yields mean MRE1:8 of 7.632% versus 22.105% for complex-tanh.Both activations are holomorphic away from their poles and omit Cauchy–Riemann penalties.Source: Table 5 · p. 15, §5.4 · p. 15

5.5 Late-training stability in 50 dimensions

  • Across five 50D checkpoints from updates 92,000–100,000, MRE1:8 ranges from 5.299% to 7.155%, with 6.367% at the prescribed endpoint.Errors fluctuate nonmonotonically within a narrow single-digit range under the fixed Degree-8 evaluation.Source: Table 6 · p. 15, §5.5 · p. 15

5.6 An exploratory 10D non-product example

  • In the exploratory 10D correlated model, HCN’s MRE1:8 is 3.684% against the pilot simulation reference, versus 19.409% for the independent-marginal comparator.The pilot reference lacks a complete discretization and sampling error bound; the independent-marginal comparator is not the correlated system’s true sum-tail reference.Source: §5.6 · pp. 15–16, Figure 3 · p. 16

6 Discussion

The discussion links HCN’s memory profile to derivative implementation and graph scheduling as well as structural holomorphy. It distinguishes approximation guarantees from unresolved optimization, inversion, and validation questions.

Structural holomorphy and memory-controlled computation

  • In the 30D profile, memory and runtime reductions reflect the combined use of native complex Cauchy subnetworks, closed-form real-axis derivatives, chunking, and sequential backpropagation.Because the tested pipelines differ in architectures, batches, derivative calculations, and graph schedules, the reported ratios do not isolate CR-penalty removal.Source: §6 · p. 17

Constrained approximation and non-product dependence

  • Under holomorphic-log assumptions, shared additive blocks approximate separable log transforms and their first derivatives, while anchored interactions approximate constrained general holomorphic targets.For correlated systems, coordinate-only representations incur irreducible uniform log-transform error because their mixed partials vanish.Source: §6 · p. 17

Separating transform approximation from numerical inversion

  • Tail error combines signed inversion and network-approximation errors, which can cancel or reinforce one another across thresholds.Exact-transform diagnostics can guide inversion-degree selection before comparing learned models under fixed evaluation rules.Source: §6 · pp. 17–18

Scope and future directions

  • Theorem 1 establishes finite-width approximation existence, but not fixed-width optimization convergence or that sampled constraints yield a globally valid probability measure.The discussion also identifies pole-contour analysis, certified larger-RBM simulation references, and single-factor resource ablations as open directions.Source: §6 · p. 18

7 Conclusion

The paper develops HCNs for BAR-based stationary-tail computation, combining holomorphic architecture constraints with memory-aware computation. It establishes constrained approximation results and reports lower errors on degree-matched comparisons, alongside scaling to a 50-dimensional benchmark.

7 Conclusion

  • HCNs combine native complex Cauchy subnetworks, exact origin anchors, boundary-coordinate exclusion, anchored interactions, and memory-aware computation for BAR-based stationary-tail estimation.The paper also establishes simultaneous value and first-derivative approximation under holomorphic-log assumptions and a mixed-derivative lower bound for coordinate-only representations under nonzero cross-covariance.Source: §7 · p. 18
  • HCN reports lower mean relative errors than NN4MGF across degree-matched 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 beyond two dimensions.Source: §7 · p. 18
  • HCN scales the 50-dimensional product-form benchmark to a single laptop GPU with stable accuracy across late-training checkpoints.The design combines closed-form real-axis derivatives, sequential branch backpropagation, and dimension-dependent batch schedules to reduce GPU memory demand and update time.Source: §7 · p. 18

Appendices

The appendices collect mathematical proofs, implementation details, diagnostics, supplementary results, and reproducibility information. Appendix A.1 derives stationary balance relations and describes boundary measures and isolation directions.

Appendices

  • The appendices cover mathematical details and constrained-approximation proofs, implementation settings, inversion diagnostics, supplementary results, and reproducibility information.These materials are organized across Appendices A–E.Source: Appendices · p. 20

A.1 Stationary balance and boundary measures

  • Stationarity and Itô’s formula yield the balance relation, while applying the same argument to coordinates gives Rm = −µ.Each boundary measure has total mass m_k and is supported on the face z_k = 0; m_k = k for the product-form family and m_k = 1 in the 2D Harrison and orthogonally reflected 10D models.Source: §A.1 · p. 20
  • Boundary-isolation directions make all boundary coefficients except the kth vanish; when R = I, the isolated argument simplifies to y_k u_k.The boundary transforms are finite-measure Laplace transforms with prescribed masses at the origin.Source: §A.1 · p. 20

A.2 Proofs of the constrained approximation results

The appendices prove value-and-derivative approximation properties for constrained Cauchy architectures, relate approximation to training losses, and document parameter allocations and analytical references.

A.2 Proofs of the constrained approximation results

  • Cauchy blocks can approximate multivariate polynomials and their first derivatives uniformly on compact sets while keeping denominators away from zero.The construction uses the rational Cauchy activation directly, with pole-free domains and polarization to represent multivariate polynomials.Source: §A.2 · pp. 20–21
  • Shared-label interpolation and origin anchoring extend the approximation argument to constrained interior and boundary targets while preserving boundary-coordinate exclusion.The proof constructs shared Cauchy blocks to approximate the interior and boundary residuals, then uses anchoring to bound value and derivative errors.Source: §A.2 · pp. 21–22
  • A coordinate-only representation has a mixed-derivative error lower bound tied to the target log transform’s cross-covariance.At the origin, the mixed derivative of the log stationary transform equals Covπ(Z_i,Z_j).Source: §A.2 · p. 22

A.3 Consequences for transforms and the training constraints

  • As the log-transform approximation tolerance tends to zero, the BAR, boundary, monotonicity, and imaginary-part losses all tend to zero.This conclusion assumes the exact transforms satisfy the BAR identity and the real-sample support lies where the target is real and coordinatewise nonincreasing.Source: §A.3 · p. 22

A.4 Parameter counts and interaction allocations

  • Coordinate and boundary embeddings with fixed widths yield 15,108 + 64d real parameters for the coordinate interior and boundary blocks.The 2D Harrison interaction architecture is allocated 15,236 parameters, matching the 2D product-form model; the 10D correlated allocation totals 23,925.Source: §A.4 · p. 23

A.5 Analytical and density-based references

  • The product-form family has independent exponential stationary coordinates with rates 1 through d, and its tail reference is given by F_d(t) = (1 − e^−t)^d.For the 2D Harrison benchmark, integrating the exact stationary density yields a one-dimensional tail integral whose quadrature supplies reference probabilities at t = 1,...,5.Source: §A.5 · p. 23

B Training and Implementation Details

The implementation uses stabilized residuals, structured sampling, sequential branch accumulation, preflight numerical checks, and fixed-degree Talbot inversion.

B.1 Stable residuals

  • The common real shift bounds each scaled BAR residual term by one in exact arithmetic while preserving the BAR equation’s zero-residual locus.At isolated boundary arguments, the shift uses the two active terms, and the pairwise boundary loss averages across samples and faces.Source: §B.1 · p. 24

B.2 Sampling and numerical settings

  • Interior batches combine nested-box samples, diagonal and near-diagonal complex inputs, and nonnegative real-diagonal inputs; branch losses use the same parameter vector before one Adam update.The interior batch assigns 50%, 20%, and 20% to the first three components, with the remainder assigned to the real diagonal.Source: §B.2 · p. 24

B.3 Preflight verification checks

  • Analytical Jacobians differ from centered finite differences by at most 10^−10 to 10^−9, while origin anchors and normal-coordinate independence hold to machine precision.Preflight checks also test denominator moduli and the finiteness of outputs, Jacobians, losses, and parameter gradients.Source: §B.3 · pp. 24–25, Table B.2 · p. 25

C.1 Evaluation implementation

  • Frozen endpoint weights and explicit degree M ∈ {5,8} are used with mpmath’s FixedTalbot inversion; neural-network queries use machine complex128.At zero, exact origin normalization bφ_0(0) = 1 supplies the removable-singularity limit.Source: §C.1 · p. 25

C.2 Signed-error decomposition

Signed-error decomposition shows how inversion and network errors can cancel or reinforce, while degree-matched comparisons show that moving from Degree 5 to Degree 8 reduces HCN errors across the reported dimensions and intervals.

C.2 Signed-error decomposition

  • At 20D and t = 8, Degree 8 yields lower total error for HCN than NN4MGF: −8.484% versus −29.028% for seed 32345.At Degree 5, opposite-signed inversion and NN4MGF network errors partially cancel; HCN’s same-signed components reinforce.Source: §C.2 · p. 26, Table C.1 · p. 26
  • At 20D and t = 8, Degree 5’s inversion error of −2.516% partially cancels NN4MGF’s +18.374% network error, producing +15.858% total error.For HCN, the same negative inversion error reinforces negative network error, yielding −18.928% and −23.220% for the two reported seeds.Source: §C.2 · p. 26, Table C.1 · p. 26

D.1 Degree sensitivity at fixed endpoints

  • Degree 8 reduces HCN MRE across all three evaluation intervals in the 20D, 30D, and 50D product-form benchmarks.The 20D, 30D, and 50D comparisons use the same trained weights within each degree pair; 20D HCN values are three-seed means.Source: §D.1 · pp. 26–27, Table D.1 · p. 27
  • For the archived 20D NN4MGF checkpoint, Degree 8 increases MRE1:8 from 8.006% to 8.148% and MRE1:10 from 16.801% to 18.908%.The passage attributes this contrast to removal of Degree-5 sign cancellation.Source: §D.1 · p. 26, Table D.1 · p. 27

D.2 Late-training diagnostics in 50 dimensions

Late-training 50D evaluations remain within reported error ranges at Degree 8. In an exploratory correlated 10D example, HCN’s sum-tail MRE is lower than the independent-marginal comparator’s, and its learned mean workload closely matches the analytical mean.

D.2 Late-training diagnostics in 50 dimensions

  • At the fixed 100,000-update endpoint, the 50D Degree-8 MRE1:8 is 6.367%.Across checkpoints 92,000–100,000, MRE1:5 ranges from 1.318% to 1.611%, while MRE1:8 ranges from 5.299% to 7.155%.Source: §D.2 · p. 27, Table D.2 · p. 27

D.3 The exploratory 10D non-product example

  • In the exploratory 10D correlated example, HCN and the independent-marginal comparator have sum-tail MREs of 3.684% and 19.409%, respectively.The learned mean total workload differs from the exact analytical mean H10 by 0.159%.Source: §D.3 · pp. 27–28, Table D.3 · p. 28

E Reproducibility and Computational Accounting

The reproducibility materials archive evaluation records and environment details. Computational accounting reports update-time and memory measurements for the controlled 30D H100 profile, alongside the cumulative base-input budget comparison.

E.1 Experimental records and provenance

  • The companion package archives pointwise predictions, hardware profiles, configurations, and SHA-256 checksums for reported benchmarks, seeds, checkpoints, and Talbot degrees.The 30D product-form experiment uses 50,000 updates with learning-rate annealing at update 40,000 under a 64M base-input budget.Source: §E.1 · p. 28

E.2 Software environment

  • Workstation and laptop runs use an NVIDIA GeForce RTX 4080 Laptop GPU, while controlled 30D resource profiles use an NVIDIA H100 NVL.The passage also lists Python 3.10.20, PyTorch 2.5.1+cu121, CUDA 12.1, and mpmath 1.3.0.Source: §E.2 · p. 28

E.3 Resource and sampling-budget accounting

  • In the 30D H100 profile, mean update times are 0.165113 s for HCN and 1.601046 s for NN4MGF.The reported means come from 200 post-warmup updates; peak allocated and reserved memory are recorded from CUDA allocator statistics.Source: §E.3 · p. 28
  • HCN’s 64M cumulative base-input budget is 0.651% of the implemented NN4MGF code-path budget.HCN uses 1,280 base inputs per update for 50,000 updates, compared with two batches of 16,384 inputs per update for NN4MGF over 300,000 updates.Source: §E.3 · pp. 28–29
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