Copula & Marginal Flows: Disentangling the Marginal from its Joint

Magnus Wiese; Ralf Korn; Robert Knobloch

arxiv: 1907.03361 · v1 · pith:BWCKIVZ5new · submitted 2019-07-07 · 💻 cs.LG · stat.ML

Copula & Marginal Flows: Disentangling the Marginal from its Joint

Magnus Wiese , Robert Knobloch , Ralf Korn This is my paper

Pith reviewed 2026-05-25 01:07 UTC · model grok-4.3

classification 💻 cs.LG stat.ML

keywords copula flowsmarginal flowsgenerative modelsnormalizing flowstail modelingdistributional propertiesdeep generative networks

0 comments

The pith

Copula and marginal flows separate dependence structure from marginal distributions to enable exact tail modeling in generative networks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard generative networks face limits on the tails they can express, with derived upper bounds showing that optimal networks often do not exist in certain Lp spaces. The paper proposes copula and marginal generative flows that model the joint dependence via a copula independent of the marginals. Once the copula approximates the uniform distribution, any desired marginal CDF can be imposed exactly. This addresses the lack of exact tail asymptotics and extrapolation in existing deep generative models such as GANs and normalizing flows. Numerical results are presented in support of the new flows.

Core claim

The central claim is that copula and marginal generative flows (CM flows) allow for an exact modeling of the tail and any prior assumption on the CDF up to an approximation of the uniform distribution, in contrast to standard generative networks whose expressible tails are bounded above.

What carries the argument

Copula and marginal generative flows (CM flows), which disentangle the dependence structure captured by the copula from the separate marginal distributions.

If this is right

Generative networks have upper bounds on the tails they can express in various situations.
In some cases no optimal generative network exists for given tail properties.
CM flows permit imposing any desired marginal CDF exactly after sufficient uniform approximation.
The approach supports extrapolation of distributional properties like tail asymptotics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The separation could improve accuracy in applications such as financial risk modeling that depend on precise tail probabilities.
The technique might be tested on synthetic data with known heavy-tailed marginals to measure extrapolation error.
It raises the question of whether similar disentangling can be applied to other generative architectures beyond flows.

Load-bearing premise

The dependence structure between variables can be fully captured by a copula that is independent of the marginal distributions.

What would settle it

Showing that a CM flow with a copula component that closely approximates the uniform distribution still cannot impose a target marginal CDF exactly would falsify the exact modeling claim.

Figures

Figures reproduced from arXiv: 1907.03361 by Magnus Wiese, Ralf Korn, Robert Knobloch.

**Figure 1.** Figure 1: (A) and (B) depict the densities f(x) of a Gaussian, Exponential, t- and Pareto distribution on the linear and log-scale respectively. it is essential to find a generative network that satisfies Fgθ,i(Z)(x) = FXi (x) (1) for all i = 1, . . . , d1 and |x| 0, since the asymptotic behavior determines the propensity to generate extremal values (see [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: (a) illustrates the theoretical density of the gumbel copula, (b) the density obtained by [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: A commutative diagram of the proposed generative flow. Following this decomposition CM flows are explicitly constructed by composing a copula flow hη : [0, 1]2 → [0, 1]2 with a marginal flow mθ : [0, 1]2 → R 2 . The marginal flow approximates the inverse CDFs F −1 X1 , F −1 X2 , whereas the copula flow approximates the generating function of C := (FX1 (X1), FX2 (X2)). Thus, a CM flow is given by gθ,η(u)… view at source ↗

**Figure 5.** Figure 5: Frank Copula 6.1 Metrics and Divergences The first performance measure we track is the Jensen-Shannon divergence (JSD) of the targeted copula C and the approximation C˜, which we denote by JSD(C k C˜). Furthermore, to assess whether the marginal distributions generated by the copula flow are uniform we approximate via Monte Carlo for i = 1, 2 and Ak = [(k − 1)/n, k/n), k = 1, . . . , n the metric T(i, n) :… view at source ↗

**Figure 6.** Figure 6: Pointwise evaluation of the JSD of the Clayton, Frank and Gumbel theoretical and copula [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

read the original abstract

Deep generative networks such as GANs and normalizing flows flourish in the context of high-dimensional tasks such as image generation. However, so far exact modeling or extrapolation of distributional properties such as the tail asymptotics generated by a generative network is not available. In this paper, we address this issue for the first time in the deep learning literature by making two novel contributions. First, we derive upper bounds for the tails that can be expressed by a generative network and demonstrate Lp-space related properties. There we show specifically that in various situations an optimal generative network does not exist. Second, we introduce and propose copula and marginal generative flows (CM flows) which allow for an exact modeling of the tail and any prior assumption on the CDF up to an approximation of the uniform distribution. Our numerical results support the use of CM flows.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

CM flows split marginals from copula dependence in normalizing flows to get exact tail control up to uniform approximation, with some non-existence bounds on standard networks.

read the letter

The paper's core move is to factor the joint via Sklar's theorem so a flow only models the copula on the unit cube while marginal CDFs are imposed exactly through inverse transforms. That lets them claim exact tail behavior once the copula part is close enough to uniform. They also derive tail upper bounds for ordinary generative networks and show that optimal networks fail to exist in some Lp settings. Both pieces are presented as new in the deep-learning literature on flows and GANs. The numerical examples are said to back the tail improvement. That is the actual content. The construction is clean on paper and directly targets a real pain point in risk-sensitive uses where standard flows smear out extremes. The non-existence claim is a useful negative result if the function-class assumptions hold. The main limitation visible from the abstract is the lack of any displayed equations, proof outlines, or quantification of the uniform approximation error. Without those, it is hard to judge how tight the tail control actually is in practice or how sensitive the method is to the quality of the copula flow. The scope stays inside normalizing flows, so broader claims about other generative models are not made. The dependence structure is assumed separable from the marginals, which is the usual copula premise but worth checking against the data in any given application. This work is aimed at people already using or extending normalizing flows who need better control over tails or prescribed marginals. A reader who cares about copula methods or extreme-value behavior in generative models will find the idea worth testing. It is coherent enough on its own terms to merit a serious referee, even if the proofs and error analysis need close scrutiny in review.

Referee Report

2 major / 1 minor

Summary. The paper claims two contributions: (1) derivation of upper bounds on tails expressible by generative networks together with Lp-space properties, including non-existence of optimal networks in various situations; (2) introduction of copula and marginal generative flows (CM flows) that model dependence on the unit cube via a copula while imposing arbitrary marginal CDFs exactly via inverse transforms, thereby achieving exact tail modeling up to the error incurred in approximating the uniform distribution. Numerical experiments are said to support the CM-flow construction.

Significance. If the tail-bound derivations and the exactness claim for CM flows hold, the work would supply a principled mechanism for controlling marginal distributions and tail asymptotics independently in deep generative models, which is relevant for risk-sensitive applications. The construction rests on the standard Sklar decomposition but applies it to flows in a way that evades the stated limitations of non-decomposed networks.

major comments (2)

[Abstract] Abstract: the derivation of upper bounds on tails and the non-existence results for optimal generative networks are asserted, yet no equations, proof sketches, or quantification of the Lp-space properties appear; without these the central claim that standard networks are provably limited cannot be verified.
[CM flows] CM-flow construction (second contribution): the assertion of 'exact modeling of the tail' up to uniform approximation requires an explicit bound showing how the copula-flow approximation error propagates into the marginal tails; absent this control the exactness claim is not established.

minor comments (1)

The abstract would be clearer if it briefly indicated the architecture or training procedure used for the numerical support of CM flows.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the two major comments point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the derivation of upper bounds on tails and the non-existence results for optimal generative networks are asserted, yet no equations, proof sketches, or quantification of the Lp-space properties appear; without these the central claim that standard networks are provably limited cannot be verified.

Authors: The abstract is a concise summary of the two contributions. The derivations of the upper bounds on tails expressible by generative networks, the Lp-space properties, and the non-existence of optimal networks in various situations are provided with equations and proof sketches in Section 3 of the manuscript, with complete proofs in the appendix. The central claims are therefore verifiable from the body of the paper rather than the abstract. revision: no
Referee: [CM flows] CM-flow construction (second contribution): the assertion of 'exact modeling of the tail' up to uniform approximation requires an explicit bound showing how the copula-flow approximation error propagates into the marginal tails; absent this control the exactness claim is not established.

Authors: In the CM-flow construction the marginal CDFs (and therefore their tails) are imposed exactly by the inverse transform applied after the copula flow; the copula approximation error affects only the dependence structure on the unit cube. The joint-tail error is consequently controlled solely by the uniform approximation error. To make this propagation explicit we will add a short lemma with the corresponding bound in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation chain consists of two independent contributions: (1) explicit upper bounds on tail behavior for standard generative networks (with proofs that optimal networks may not exist in certain Lp settings), and (2) a new CM-flow construction that applies Sklar's theorem to factor the joint into a copula component modeled by a flow on the unit cube plus exact marginal CDFs imposed via inverse transforms. Neither step reduces its claimed result to a fitted parameter, self-citation, or definitional renaming; the 'exact marginal' property holds by the explicit architectural separation rather than by construction from the target quantity itself. No load-bearing self-citations or ansatz smuggling appear in the stated claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard copula theory (domain assumption) and the modeling choice that marginals and dependence can be separated without loss of expressivity for tail purposes (ad_hoc_to_paper). No free parameters or invented entities with independent evidence are described in the abstract.

axioms (1)

domain assumption Marginal distributions and dependence structure can be modeled independently via copulas without affecting tail asymptotics.
Invoked to justify the CM-flow split in the second contribution.

invented entities (1)

Copula and marginal generative flows (CM flows) no independent evidence
purpose: Separate modeling of marginal CDFs (including tails) from joint dependence.
Newly proposed architecture; no external falsifiable evidence supplied in abstract.

pith-pipeline@v0.9.0 · 5668 in / 1322 out tokens · 20866 ms · 2026-05-25T01:07:10.382868+00:00 · methodology

discussion (0)

Forward citations

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Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · cited by 2 Pith papers · 5 internal anchors

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Pair-Copula Constructions of Multiple Dependence

Kjersti Aas et al. “Pair-Copula Constructions of Multiple Dependence”. In: Insurance: Mathe- matics and Economics 44 (Apr. 2009), pp. 182–198

work page 2009

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Towards Principled Methods for Training Generative Adversarial Networks

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work page internal anchor Pith review Pith/arXiv arXiv 2017

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Size-Noise Tradeoffs in Generative Networks

Bolton Bailey and Matus J Telgarsky. “Size-Noise Tradeoffs in Generative Networks”. In: Advances in Neural Information Processing Systems 31 . Ed. by S. Bengio et al. Curran Associates, Inc., 2018, pp. 6490–6500

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Vines - A new graphical model for dependent random variables

Tim Bedford and Roger Cooke. “Vines - A new graphical model for dependent random variables”. In: Annals of Statistics 30 (Sept. 1999)

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Large Scale GAN Training for High Fidelity Natural Image Synthesis

Andrew Brock, Jeff Donahue, and Karen Simonyan. “Large Scale GAN Training for High Fidelity Natural Image Synthesis”. In: CoRR abs/1809.11096 (2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018

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Extreme value theory: an introduction

Laurens De Haan and Ana Ferreira. Extreme value theory: an introduction. Springer Science & Business Media, 2007

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NICE: Non-linear Independent Compo- nents Estimation

Laurent Dinh, David Krueger, and Yoshua Bengio. “NICE: Non-linear Independent Compo- nents Estimation”. In: 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Workshop Track Proceedings. 2015

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Density estimation using Real NVP

Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. “Density estimation using Real NVP”. In: CoRR abs/1605.08803 (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016

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Copula Bayesian Networks

Gal Elidan. “Copula Bayesian Networks”. In: Advances in Neural Information Processing Systems 23. Ed. by J. D. Lafferty et al. Curran Associates, Inc., 2010, pp. 559–567

work page 2010

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Practical Extreme Value Modelling of Hydrological Floods and Droughts: A Case Study

Kolbjørn Engeland, Hege Hisdal, and Arnoldo Frigessi. “Practical Extreme Value Modelling of Hydrological Floods and Droughts: A Case Study”. In: Extremes 7 (Mar. 2004), pp. 5–30

work page 2004

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Maxout Networks

Ian J. Goodfellow et al. “Maxout Networks”. In: Proceedings of the 30th International Confer- ence on International Conference on Machine Learning - Volume 28. ICML’13. Atlanta, GA, USA: JMLR.org, 2013, pp. III-1319–III-1327

work page 2013

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Generative Adversarial Nets

Ian Goodfellow et al. “Generative Adversarial Nets”. In: Advances in Neural Information Processing Systems 27. Ed. by Z. Ghahramani et al. Curran Associates, Inc., 2014, pp. 2672– 2680

work page 2014

[14] [14]

Delving Deep into Rectiﬁers: Surpassing Human-Level Performance on ImageNet Classiﬁcation

Kaiming He et al. “Delving Deep into Rectiﬁers: Surpassing Human-Level Performance on ImageNet Classiﬁcation”. In: Proceedings of the 2015 IEEE International Conference on Computer Vision (ICCV). ICCV ’15. Washington, DC, USA: IEEE Computer Society, 2015, pp. 1026–1034. ISBN : 978-1-4673-8391-2

work page 2015

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Approximation capabilities of multilayer feedforward networks

Kurt Hornik. “Approximation capabilities of multilayer feedforward networks”. In: Neural Networks 4.2 (1991), pp. 251–257. ISSN : 0893-6080

work page 1991

[16] [16]

Neural Autoregressive Flows

Chin-Wei Huang et al. “Neural Autoregressive Flows”. In: CoRR abs/1804.00779 (2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018

[17] [17]

Monte Carlo Methods and Models in Finance and Insurance

Ralf Korn, Elke Korn, and Gerald Kroisandt. “Monte Carlo Methods and Models in Finance and Insurance”. In: (Jan. 2010)

work page 2010

[18] [18]

Introduction to Vine Copulas

Nicole Krämer and Ulf Schepsmeier. Introduction to Vine Copulas. 2011

work page 2011

[19] [19]

Efﬁcient BackProp

Yann LeCun et al. “Efﬁcient BackProp”. In: Neural Networks: Tricks of the Trade, This Book is an Outgrowth of a 1996 NIPS Workshop. London, UK, UK: Springer-Verlag, 1998, pp. 9–50. ISBN : 3-540-65311-2. 9

work page 1996

[20] [20]

Which Training Methods for GANs do actually Converge?

Lars M. Mescheder. “On the convergence properties of GAN training”. In: CoRR abs/1801.04406 (2018)

work page internal anchor Pith review Pith/arXiv arXiv 2018

[21] [21]

Rectiﬁed Linear Units Improve Restricted Boltzmann Machines

Vinod Nair and Geoffrey E. Hinton. “Rectiﬁed Linear Units Improve Restricted Boltzmann Machines”. In: Proceedings of the 27th International Conference on International Conference on Machine Learning. ICML’10. Haifa, Israel: Omnipress, 2010, pp. 807–814.ISBN : 978-1- 60558-907-7

work page 2010

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The Double Pareto-Lognormal Distribution—A New Parametric Model for Size Distributions

William Reed and Murray Jorgensen. “The Double Pareto-Lognormal Distribution—A New Parametric Model for Size Distributions”. In: Communications in Statistics. Theory and Methods 8 (May 2004)

work page 2004

[23] [23]

Variational Inference with Normalizing Flows

Danilo Jimenez Rezende and Shakir Mohamed. “Variational Inference with Normalizing Flows”. In: Proceedings of the 32Nd International Conference on International Conference on Machine Learning - Volume 37. ICML’15. Lille, France: JMLR.org, 2015, pp. 1530–1538

work page 2015

[24] [24]

Survival Probabilities Based on Pareto Claim Distributions

Hilary L. Seal. “Survival Probabilities Based on Pareto Claim Distributions”. In: ASTIN Bulletin 11.1 (1980), pp. 61–71. 10 A Proofs Proof of Lemma 6. P  a d0∑ j=1 Zj +b>x   = 1− P  a d0∑ j=1 Zj≤x−b   ≤ 1− P   d0⋂ j=1 { aZj≤ x−b d0 }  = P   d0⋃ j=1 { aZj > x−b d0 }  ≤d0 P ( aZ 1 > x−b d0 ) =d0 P (ad 0Z1 +b>x ). Proof of Corollary 7. SinceX...

work page 1980