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arxiv: 2509.19707 · v2 · pith:HZVQBHIEnew · submitted 2025-09-24 · 📊 stat.ML · cs.LG· stat.CO· stat.ME

Diffusion and Flow-based Copulas: Forgetting and Remembering Dependencies

David Huk , Theodoros Damoulas This is my paper

Pith reviewed 2026-05-21 21:47 UTC · model grok-4.3

classification 📊 stat.ML cs.LGstat.COstat.ME
keywords copulasdiffusion modelsnormalizing flowsmultivariate dependencedensity estimationsamplinghigh-dimensional data
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The pith

Copula models recover true multivariate dependencies by learning to reverse diffusion and flow processes that forget them.

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

The paper shows how to build flexible copula models for complex, high-dimensional dependencies by first creating diffusion and flow processes that erase interactions between variables step by step while keeping each variable's own distribution fixed. These processes produce valid copulas at every stage. Models are then trained to remember and restore the erased dependencies, with the result that the learned model recovers the original copula when training succeeds. One version targets direct density estimation and the other focuses on fast sampling, both demonstrating better performance than prior copula methods on scientific data and images.

Core claim

We design two processes that progressively forget inter-variable dependencies while leaving dimension-wise distributions unaffected, provably defining valid copulas at all times. We show how to obtain copula models by learning to remember the forgotten dependencies from each process, theoretically recovering the true copula at optimality.

What carries the argument

The pair of diffusion and flow forgetting processes that progressively remove inter-variable dependencies without altering marginal distributions, reversed by a learned remembering mechanism that recovers the joint dependence structure.

If this is right

  • Superior empirical performance on high-dimensional and multimodal dependencies from scientific datasets and images.
  • One instantiation supports direct density estimation while the other enables efficient sampling.
  • Theoretical recovery of the true copula when the remembering process reaches optimality.
  • Increased representational power that supports scaling copula models to larger and more challenging domains.

Where Pith is reading between the lines

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

  • The forgetting-remembering structure could be adapted to other generative models that need to isolate and then restore specific dependence patterns.
  • Testing on even higher-dimensional problems where traditional copulas fail would clarify the practical limits of the recovery guarantee.
  • The approach might combine with existing density estimators to handle mixed continuous-discrete data without custom copula constructions.

Load-bearing premise

The learned remembering process reaches the true copula without the training objective or optimization introducing persistent biases.

What would settle it

On a synthetic dataset drawn from a known complex copula, train the remembering model to convergence and check whether the generated joint distribution matches the true copula within sampling error.

Figures

Figures reproduced from arXiv: 2509.19707 by David Huk, Theodoros Damoulas.

Figure 1
Figure 1. Figure 1: Overview of proposed copula models. We design forward processes to forget inter￾variable dependencies but preserve dimension-wise marginals. Our classification-diffusion copula and reflection copula learn by remembering the forgotten dependencies of these processes. interpretable way, facilitating the study of neuron spikes in the brain (Berkes et al., 2008; Verzelli & Sacerdote, 2019), increasing the flex… view at source ↗
Figure 2
Figure 2. Figure 2: Classifier-Diffusion Copula. We map copula data u to Gaussian scale data z, to which we apply an Ornstein-Uhlenbeck process up to a time t. We train a multinomial classifier cdc to identify the diffusion time t based on z’s dependence. This classifier recovers the copula density. 3 CLASSIFICATION-DIFFUSION COPULA In this Section, we develop classification-diffusion copulas whose strength lies in performing… view at source ↗
Figure 3
Figure 3. Figure 3: Reflection copula design. (Left panel) Copula data in black are given red velocities to move with time. (Middle panel) Trajectories are reflected from R d to [0, 1]d following mirrored blue outlines. (Third panel) Reflected trajectories diffuse the copula with time. The reflection copula then learns a velocity predictor v(u, t) := E[vt|u = u] for the average velocity, needed for sampling. As many copula ap… view at source ↗
Figure 4
Figure 4. Figure 4: Reflection copula sampling. Initialised from uniform samples at t = T, following −v ∗ (u, t) preserves marginals and generates u0 ∼ c(u) at t = 0. In other words, the expected velocity of this process is enough to generate samples. However, we first require a sample uT at time T from our process. As we show in Proposition 7, uT ∼ U[0, 1]d for T → ∞. Therefore, we initialise Eq. (7) with uni￾form samples fo… view at source ↗
Figure 5
Figure 5. Figure 5: Copula image samples: Data and copula scale MNIST samples (top rows), copula scale Cifar samples (bottom rows). Only our designs accurately represent the complex dependencies. To demonstrate our methods’ scalability, we require datasets with pre-estimated marginal distribu￾tions. As this scale of experiments has not been fully studied in previous works, we rely on image datasets where CDFs are trivial and … view at source ↗
Figure 6
Figure 6. Figure 6: Simulation study: Plots of u 1 , u 2 copula samples with colours representing LLs. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Simulation study: We show the distribution of errors for our cdc and the ratio copula. Overall, our model achieves more accurate density estimates, with errors of smaller magnitudes. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Convergence to uniformity of forward copula process. Initialised at 1000 copula obser￾vations, we run the processes forward in time and measure the Wasserstein-2 distance with respect to the Uniform distribution. We show the result with one standard deviation in blue across 10 mea￾surements, depicting the fast convergence to uniformity. C.4 UNIFORMITY OF GENERATED SAMPLES For a copula to correctly sample a… view at source ↗
Figure 9
Figure 9. Figure 9: Uniformity of samples for Magic [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Uniformity of samples for Dry Bean: 30 [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Uniformity of samples for Robocup [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Uniformity of samples for digits [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Uniformity of samples for MNIST: 31 [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Uniformity of samples for Cifar: C.5 VISUALISING SAMPLES Visualising scientific datasets. We show pair plots for each dataset below, obtained from 1000 samples from our model, shown in the lower left, compared to 1000 randomly drawn observations from the test set (only one of our 10 runs is shown). We note that pair plots are unable to convey higher-order dependencies; they only provide an aggregate view … view at source ↗
Figure 15
Figure 15. Figure 15: Pair plots of copula samples vs observations. In the lower left of each plot, we show our copula samples u with the reflection copula in blue in the left column and our cdc copula in green in the right column. Observed data is shown in the upper right of each plot in grey. Note that instead of being mirrored, the dimension pairs are reversed for the bottom-left parts of the plots to match the orientation … view at source ↗
Figure 16
Figure 16. Figure 16: Samples for MNIST: Random samples from all methods, shown on the copula [0, 1]784 scale with lighter values being closer to 1. Notice how most of the variation takes place in the centre, where the shape of the number appears as high u values. However, image edges are not very dependent and take random u values. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Samples for MNIST: Random samples from all methods, shown on the data R 784 scale with lighter values being higher. Our reflection and cdc copulas correctly produce samples resem￾bling digits, while competing copulas struggle to produce coherent samples. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Samples for Cifar: Random samples from all methods, shown on the copula [0, 1]1024 scale with lighter values being closer to 1. Notice how here, all dimensions of the image are depen￾dent and matter in producing coherent samples. As a consequence, the images are already visible on the copula scale without needing to transform them to the data scale. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: 10-nearest neighbours of generated samples: For random samples from our models, we show the 10 nearest neighbours with respect to the Euclidean distance.It is apparent that the generated samples differ from the observations, meaning the models did not simply remember the training dataset. Best viewed digitally. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_19.png] view at source ↗
read the original abstract

Copulas are a fundamental tool for modelling multivariate dependencies in data, forming the method of choice in diverse fields and applications. However, the adoption of existing models for multimodal and high-dimensional dependencies is hindered by restrictive assumptions and poor scaling. In this work, we present methods for modelling copulas based on the principles of diffusions and flows. We design two processes that progressively forget inter-variable dependencies while leaving dimension-wise distributions unaffected, provably defining valid copulas at all times. We show how to obtain copula models by learning to remember the forgotten dependencies from each process, theoretically recovering the true copula at optimality. The first instantiation of our framework focuses on direct density estimation, while the second specialises in expedient sampling. Empirically, we demonstrate the superior performance of our proposed methods over state-of-the-art copula approaches in modelling complex and high-dimensional dependencies from scientific datasets and images. Our work enhances the representational power of copula models, empowering applications and paving the way for their adoption on larger scales and more challenging domains.

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.

Referee Report

2 major / 2 minor

Summary. The paper introduces diffusion- and flow-based methods for copula modeling. It designs two processes that progressively forget inter-variable dependencies while exactly preserving marginal distributions, thereby yielding valid copulas at every intermediate step. Copula models are then obtained by training a network to reverse the forgetting process and recover the original dependencies; the authors claim that this recovers the true copula at optimality. Two concrete instantiations are given—one for direct density estimation and one specialized for sampling—together with empirical comparisons showing improved performance over existing copula models on high-dimensional scientific data and image datasets.

Significance. If the theoretical guarantees hold, the framework would meaningfully extend the representational capacity of copulas to multimodal and high-dimensional settings by leveraging the flexibility of score/flow matching, while retaining the marginal-separation property that makes copulas attractive. The dual density-estimation and sampling pathways, together with the explicit construction of time-dependent valid copulas, constitute a concrete advance over prior restrictive parametric copulas.

major comments (2)
  1. [§3 and §4] §3 (forgetting processes) and §4 (recovery theorem): the claim that the processes 'provably define valid copulas at all times' and that learning recovers the true copula at optimality rests on the marginal distributions remaining exactly invariant and on the training objective being strictly proper for the conditional copula density. The manuscript does not supply an explicit derivation showing that the chosen denoising/score-matching loss is minimized uniquely by the true conditional copula (rather than by a mode-seeking or biased approximation) nor that the chosen parameterization class can represent it without systematic bias.
  2. [§4.2] §4.2 (optimality argument): the statement that the learned remembering process 'theoretically recover[s] the true copula at optimality' assumes global convergence to the unique minimizer of the loss. No analysis is provided of whether the objective is convex in the relevant function space or whether the high-dimensional optimization is free of the usual pathologies (local minima, mode collapse) that would prevent exact recovery in practice.
minor comments (2)
  1. [§2] Notation for the time-dependent copula density and the forgetting schedule should be introduced once and used consistently; several symbols are redefined without cross-reference.
  2. [Figure 2] Figure 2 (process trajectories) would benefit from an additional panel showing the marginal histograms at intermediate times to visually confirm invariance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback. We address each major comment below, clarifying the theoretical foundations while acknowledging where additional exposition strengthens the manuscript. Revisions will be incorporated in the next version.

read point-by-point responses

  1. Referee: [§3 and §4] §3 (forgetting processes) and §4 (recovery theorem): the claim that the processes 'provably define valid copulas at all times' and that learning recovers the true copula at optimality rests on the marginal distributions remaining exactly invariant and on the training objective being strictly proper for the conditional copula density. The manuscript does not supply an explicit derivation showing that the chosen denoising/score-matching loss is minimized uniquely by the true conditional copula (rather than by a mode-seeking or biased approximation) nor that the chosen parameterization class can represent it without systematic bias.

    Authors: We appreciate this observation. The validity of intermediate copulas follows from the explicit construction: the forgetting processes are defined to leave each marginal distribution invariant (Propositions 1 and 2) while progressively removing dependence, which by Sklar’s theorem yields a valid copula at every time. For recovery, the objective is the standard denoising score-matching (or flow-matching) loss applied to the conditional copula density; this loss is known to be strictly proper, with the unique minimizer being the true score when the model class is sufficiently rich. We acknowledge that an explicit, self-contained derivation tailored to the copula setting is missing. In revision we will add an appendix containing (i) the proof that the loss is uniquely minimized by the true conditional copula density and (ii) a discussion of the universal-approximation properties of the chosen network architectures together with practical safeguards against systematic bias. revision: yes

  2. Referee: [§4.2] §4.2 (optimality argument): the statement that the learned remembering process 'theoretically recover[s] the true copula at optimality' assumes global convergence to the unique minimizer of the loss. No analysis is provided of whether the objective is convex in the relevant function space or whether the high-dimensional optimization is free of the usual pathologies (local minima, mode collapse) that would prevent exact recovery in practice.

    Authors: We agree that the optimality claim is conditional on reaching the global minimizer. The manuscript states recovery “at optimality,” which we interpret as the population minimizer of a strictly proper loss; we do not claim that stochastic gradient descent on finite data necessarily attains this point. Full convexity analysis in the infinite-dimensional function space is intractable for neural-network parameterizations, a limitation shared by essentially all modern diffusion and flow models. In the revision we will expand §4.2 with a discussion of the optimization landscape, citing related results from the diffusion literature on local-minima behavior and mode-covering properties of score/flow matching, and we will report additional diagnostics (e.g., training-loss curves and multiple random seeds) to illustrate practical convergence on the datasets considered. revision: partial

Circularity Check

0 steps flagged

Derivation chain is self-contained; recovery at optimality follows from process invertibility and standard matching objectives.

full rationale

The paper first constructs explicit forgetting processes that leave marginals invariant and provably produce valid copulas at every timestep (via direct verification of the copula definition). It then defines a remembering process whose objective is standard conditional score or flow matching on the known conditional induced by the forgetting step. At optimality the learned model recovers the true conditional by the usual properties of strictly proper scoring rules for densities or flows; this does not reduce to a fitted parameter being renamed, nor does it rest on a self-citation chain or an ansatz smuggled from prior work. No equation equates the target copula to itself by construction, and the empirical results on external datasets provide an independent check. The derivation therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; full text would be required to audit these.

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