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Multifidelity Formulations for Triangular Transport

Owen Davis, Daniel Sharp, Youssef Marzouk, Gianluca Geraci

arXiv:2610.00698v1stat.COcs.NAmath.NAstat.ML

TL;DR

Learning triangular transport maps from samples is difficult when high-fidelity data are scarce. The paper proposes hierarchical map composition and non-hierarchical corrections that use more abundant low-fidelity data. Across several numerical examples, the value of multifidelity information depends on whether low-fidelity structure reduces the complexity to be learned from scarce high-fidelity samples.

  • Problem

    Accurate triangular transport maps can require many target samples, which may be difficult to obtain for expensive simulations and physical experiments.

    Source: §1 · pp. 1–2

  • Method

    The paper combines scarce high-fidelity samples with more plentiful low-fidelity samples using either composed maps along a fidelity hierarchy or corrections to pretrained lower-fidelity maps.

    Source: §1 · p. 2, §6 · p. 23

  • Results

    Across polynomial verification, Neal’s funnel, and Lotka–Volterra simulation-based inference, multifidelity information is valuable when lower-fidelity structure reduces the complexity learned from scarce high-fidelity samples.

    Source: §6 · p. 23

  • Takeaways & Limitations

    Non-hierarchical corrections can suppress less useful sources, while hierarchical methods depend directly on the fidelity ordering.

    Source: §6 · p. 23

    Hierarchical methods assume each staged map is simpler than the direct high-fidelity map, a condition described as natural for nested model families.

    Source: §4.3 · p. 10

Abstract

from arXiv · show

We develop multifidelity methods for constructing triangular transport maps from samples, when high-fidelity data are scarce but lower-fidelity data are more abundant. Using this set of multifidelity data, we approximate a triangular transport map that bijectively maps between a tractable reference density and the high-fidelity target distribution. We introduce two strategies to leverage low-fidelity data: a hierarchical approach that composes maps between adjacent fidelity levels, and a non-hierarchical method that incorporates low-fidelity information through monotonicity-preserving corrections to the map parameterization. Numerical experiments compare these strategies with single-fidelity transport and demonstrate how the proposed multifidelity approaches can improve map estimation from limited high-fidelity data. To illustrate the broader utility of the learned maps, we also deploy them in a downstream amortized simulation-based inference task. This example shows that multifidelity improvements in map estimation can translate to improved conditional sampling and uncertainty quantification when high-fidelity data are scarce.

1 Introduction

Learning accurate triangular transport maps from target samples can require many samples, making construction difficult when high-fidelity simulations or experiments are costly. The paper combines scarce high-fidelity data with more abundant lower-fidelity data through hierarchical map composition or non-hierarchical corrections, and evaluates these approaches on distribution-learning and simulation-based inference tasks.

1 Introduction

  • Accurate map construction can require many target samples, which is challenging when simulations or physical experiments are expensive.The paper addresses this data-scarcity problem using multifidelity information from sources with different costs and accuracies.Source: §1 · pp. 1–2
  • The methods assume scarce high-fidelity samples and more plentiful samples from related lower-fidelity distributions to estimate a map coupling the high-fidelity target and a tractable reference.The resulting map supports sampling, density evaluation, and simulation-based inference.Source: §1 · p. 2
  • The hierarchical approach composes maps between adjacent fidelity levels, while the non-hierarchical approach uses corrections to incorporate low-fidelity sources without requiring an ordering.The non-hierarchical construction can automatically suppress misleading lower-fidelity information; hierarchical methods use fixed-reference or changing-reference variants.Source: §1 · p. 2
  • The paper evaluates both method families on distribution-learning tasks and a canonical simulation-based inference problem adapted to multifidelity data.The paper positions these approaches against existing multifidelity settings where dependency structures can affect performance, including hierarchies vulnerable to error propagation from uninformative or corrupted sources.Source: §1 · p. 2

2 Related work

Prior multifidelity inference methods use multiple-fidelity information for posterior estimation, sampling, or transport-assisted inference, but do not train a high-fidelity triangular transport map directly from samples across fidelity levels. The closest related work addresses narrower spatial or Bayesian-inference settings, while this paper focuses on direct map design and also considers settings without a hierarchy.

2 Related work

  • Few related methods use the full multifidelity dataset to learn a transport map itself, and fewer support conditional sampling, a key feature of triangular transport.The paper distinguishes its focus from multifidelity methods more broadly.Source: §2 · p. 3
  • Multifidelity simulation-based inference methods [28] [50] characterize posteriors but do not develop triangular transport constructions.[49, 50] use multifidelity approximate Bayesian computation, while [28] applies transfer learning to a neural density estimator.Source: §2 · p. 3
  • Likelihood-based methods [39] [45] use multifidelity information with sampling or transport, but do not train a high-fidelity triangular map from samples across fidelity levels.[39] couples MCMC chains across fidelity levels, uses triangular maps to accelerate that strategy, and [45] uses a low-fidelity map to guide high-fidelity proposals.Source: §2 · p. 3
  • The closest related works address spatial multiscale generation [8] or Bayesian inference [56], whereas this paper studies direct map estimation from available samples and also examines cases without a hierarchy.The paper is not restricted to spatial fields, resolution hierarchies, or conditional generation from paired coarse-fine data.Source: §2 · p. 3

3 Triangular transport

The paper uses monotone triangular maps to couple target and reference distributions, learns them from target samples with a sample-based objective, and enforces monotonicity through an integrated-rectifier parameterization.

3 Triangular transport

  • A triangular map has kth output depending only on the first k inputs, and monotonicity in the kth input makes the map invertible.The map pushes the target density to the reference; applying its inverse to reference samples yields target samples.Source: §3 · pp. 3–4

3.1 Learning a transport map from samples

  • The learning objective approximates the target-to-reference map from target samples, with a sample-average form linked to maximum likelihood estimation.With a standard Gaussian reference, the objective separates across triangular components, allowing parallel training.Source: §3.1 · p. 4

3.2 Map component parameterization

  • An integrated-rectifier parameterization enforces monotonicity by construction through a positive rectifier applied to each component.The paper chooses SoftPlus to satisfy rectifier-specific conditions discussed for the reparameterized optimization.Source: §3.2 · pp. 4–5

4 Triangular transport from multifidelity information

The section develops hierarchical and non-hierarchical ways to use abundant low-fidelity samples to reduce what must be learned from scarce high-fidelity data. The methods differ in whether they compose maps along a fidelity ordering or combine peer sources in one corrected parameterization.

4 Triangular transport from multifidelity information

  • The two method families target reduced learning complexity: hierarchical methods compose adjacent-fidelity maps, while the non-hierarchical method corrects the high-fidelity parameterization using all lower-fidelity sources.The framework assumes fewer high-fidelity samples than lower-fidelity samples and seeks to improve on high-fidelity-only estimation.Source: §4 · p. 5

4.1 Hierarchical multifidelity transport

  • Hierarchical transport assumes an ordering of target densities and composes maps between adjacent fidelity levels, using either changing or fixed references.The two formulations trade adjacent-fidelity transport for preservation of a common reference structure.Source: §4.1 · p. 6

4.1.1 Changing-reference hierarchical method

  • Changing-reference training uses map-induced approximations of lower-fidelity targets as later-stage references, which may lose the convexity and componentwise separability available with a Gaussian reference.The construction learns stages sequentially, with each induced reference used at the next higher-fidelity stage.Source: §4.1.1 · pp. 6–7

4.1.2 Fixed-reference hierarchical composition

  • Fixed-reference training keeps the prescribed reference at every stage and transforms each stage’s samples by the lower-fidelity map composition, preserving componentwise separability for a Gaussian reference.When the hierarchy is informative, each higher-fidelity stage can learn a residual correction.Source: §4.1.2 · pp. 7–8

4.2 Non-hierarchical multifidelity transport

  • The non-hierarchical method treats lower-fidelity densities as unordered peers and combines corrected low-fidelity map components with a high-fidelity shift.It assumes the high-fidelity map shares useful structure with at least some low-fidelity maps.Source: §4.2 · p. 8, §4.3 · p. 11

4.2.1 Formulation

  • Its monotonicity-preserving scale-shift construction separately scales and shifts nonmonotone and monotone map components, while trainable corrections adapt pretrained low-fidelity maps.The monotone linear combination is placed inside the rectifier to preserve monotonicity.Source: §4.2.1 · p. 9

4.2.2 Training

  • Training first pretrains low-fidelity maps independently, then jointly fits corrections and the high-fidelity map, with low-fidelity losses acting as a multifidelity regularizer.Trainable scales can suppress low-fidelity contributions that are not useful for approximating the high-fidelity map.Source: §4.2.2 · pp. 9–10

4.3 Discussion and comparison

  • The methods rely on different structural assumptions: hierarchical maps must simplify along an ordering, while the peer method requires its corrected scale-shift representation to approximate the high-fidelity map.The non-hierarchical construction is tied to triangular parameterizations with componentwise monotonicity-preserving structure, although the broader peer-source idea is not limited to triangular maps.Source: §4.3 · pp. 10–11

5 Numerical results

The numerical study evaluates multifidelity triangular transport maps on targets with known high-fidelity maps and in conditional inference with scarce high-fidelity simulations. It examines map-estimation results and downstream posterior accuracy across the methods.

5 Numerical results

  • The section compares single-fidelity transport with hierarchical and non-hierarchical multifidelity methods across two known-map estimation tasks and a Lotka–Volterra inference task.For the first two tasks, accuracy is measured against the known map; the final task considers conditional inference where the high-fidelity map is unavailable.Source: §5 · p. 11

5.1 Simple analytical verification example; 5.2 Softened Neal’s funnels

  • Across both map-estimation examples, NH with HF-TO-1 has the best aggregate performance; H-FR can also perform well under favorable ordering.In the first example, H-FR with a favorable hierarchy and TO-4 is similarly effective; in the funnel example, H-FR is effective only when the hierarchy aligns with the target geometry.Source: Figure 4 · p. 14, §5.1 · p. 15, Figure 7 · p. 17

5.1 Simple analytical verification example

  • In the 2D example, NH with HF-TO-1 most clearly improves on the single-fidelity TO-4 baseline, while hierarchical performance depends strongly on ordering.Placing the more informative 𝜋2 directly below 𝜋0 helps the hierarchical methods; the changing-reference method costs substantially more to train without a commensurate accuracy advantage.Source: §5.1 · pp. 14–16

5.2 Softened Neal’s funnels

  • In the 5D funnel example, H-FR performs worse than SF and NH under the misaligned hierarchy but can reduce error under the favorable ordering; NH’s clearest, most stable gains appear with HF-TO-1.Restricting the high-fidelity shift lets corrected low-fidelity components and trainable scales supply nonlinear structure; NH’s aggregate allocation sensitivity is modest, while H-FR remains strongly hierarchy-dependent.Source: §5.2 · pp. 17–18

5.3 Multifidelity simulation-based inference

  • The Lotka–Volterra example uses Gillespie SSA for high fidelity, tau-leaping for one lower fidelity, and a deterministic mean-field ODE for the other.Because SSA produces stochastic output for a fixed input, the example is a likelihood-free inference problem without reference samples absent methods such as ABC.Source: §5.3 · p. 19
  • The study varies high-fidelity sample size across N0 = 100, 500, 1000, and 2000 while keeping lower-fidelity sample pools fixed, and compares single-fidelity TO-3 with multifidelity TO-3 and HF-TO-1 maps.Because the high-fidelity KR map is unknown, amortized inference accuracy is evaluated with prior-predictive risk metrics.Source: §5.3 · p. 20
  • The non-hierarchical HF-TO-1 map gives the clearest and most consistent improvement over the single-fidelity TO-3 baseline when high-fidelity data are scarce.This advantage is strongest for posterior mean and MMD risks.Source: Figure 10 · p. 21, §5.3 · p. 21
  • NH HF-TO-1 improves Förstner risk over the baseline only at the smallest N0, then stagnates above baseline as N0 increases.The low-order high-fidelity shift can allocate posterior mass to correct regions and reduce MMD while limiting recovered covariance structure; NH TO-3 does not show this large-N0 stagnation.Source: Figure 10 · p. 21, §5.3 · p. 21
  • Neither H-FR lower-fidelity hierarchy convincingly outperforms single-fidelity transport across all three risks, and H-FR generally performs better with a TO-3 than HF-TO-1 final stage.Dflf generally outperforms Cflf, especially with HF-TO-1; the natural ODE-to-tau-leaping-to-SSA simulator order may not yield the simplest transport hierarchy.Source: Figure 10 · p. 21, §5.3 · pp. 21–22
  • In paired trials using identical high-fidelity training samples, NH HF-TO-1 consistently improves posterior mean and MMD risks in sparse-data trials, with its advantage shrinking as N0 increases.Figure 11 compares multifidelity-to-baseline error ratios paired by training replicate.Source: Figure 11 · p. 22, §5.3 · p. 22
  • At N0 = 100, the displayed NH HF-TO-1 posterior agrees most credibly with the SMC-ABC reference, while the single-fidelity posterior is more diffuse and H-FR TO-3 marginals vary.This single conditional example is qualitative and should not be overinterpreted; comparison maps use identical high-fidelity training samples.Source: §5.3 · pp. 22–23, Figure 12 · p. 23

6 Conclusion

The conclusion presents hierarchical and non-hierarchical multifidelity triangular transport as ways to use abundant lower-fidelity data when high-fidelity samples are scarce. Their benefit depends on whether lower-fidelity structure reduces the complexity learned from high-fidelity samples.

6 Conclusion

  • Across the polynomial, Neal’s funnel, and Lotka–Volterra examples, multifidelity information helps when it reduces the map complexity that scarce high-fidelity samples must resolve.Its value is not automatic.Source: §6 · p. 23
  • The non-hierarchical method works best when its parameterization encourages use of pretrained low-fidelity maps, leaving high-fidelity samples to learn low-complexity corrections and scale coefficients.The scale coefficients can suppress less useful information sources, making this method insensitive to the chosen fidelity hierarchy.Source: §6 · p. 23
  • Hierarchical methods depend directly on fidelity ordering: favorable orderings can simplify the final-stage map, while poor ones propagate errors and make later stages harder to learn.A favorable ordering can also make the method robust to a naïve final-stage parameterization.Source: §6 · p. 23
  • The fixed-reference variant is generally faster and more stable to train than the changing-reference variant.The changing-reference variant discards the standard Gaussian reference after its first stage.Source: §6 · p. 24
  • Future work includes cost-aware adaptive allocations and parameterizations, alternative bases, unnormalized-density inputs, and federated settings.Hierarchical approaches transfer readily to other transport representations, whereas non-hierarchical approaches for those methods remain an open direction.Source: §6 · p. 24

Appendices

The appendix supplies implementation details for triangular transport, including inversion, map parameterization, and optimization.

A Additional Details on Triangular Transport

A.1 Map inversion

  • Inverting a monotone triangular map proceeds recursively, requiring d scalar root-finding problems, each with a unique solution when ∂_k S_k > 0.Earlier coordinates are computed before solving the next one-dimensional problem.Source: §A.1 · p. 29

A.2 Additional information about map component parameterization

  • Map components use finite expansions over multi-indices, with total-order index sets defining the map parameterization convention.The expansion uses tensor products of one-dimensional basis functions.Source: §A.2 · p. 29
  • Hermite functions are used for terms of order two and higher, while constant and linear terms retain standard Hermite polynomials.The Hermite-function damping parameter is fixed at 30; it improves stability outside the training region, though excessive damping can reduce extrapolation accuracy.Source: §A.2 · pp. 29–30
  • The softplus rectifier is smooth and strictly positive, enforcing monotonicity through the map parameterization.Different rectifiers can affect parameter identifiability and optimization behavior.Source: §A.2 · p. 30

A.3 Optimization details

  • BFGS and L-BFGS routines implemented in SciPy solve the finite-dimensional training problems for all three map approaches.Source: §A.3 · p. 30

B Hierarchical Transport and Deep Maps

The appendix relates hierarchical multifidelity constructions to deep transport maps by describing compositions of triangular maps.

B Hierarchical Transport and Deep Maps

  • Composing maps in coarse approximation spaces can improve transport approximations without substantially enriching each individual map parameterization.The appendix relates hierarchical constructions to deep-map approaches, including changing-reference and fixed-reference composition directions.Source: B Hierarchical Transport and Deep Maps · p. 30

B.1 Single-fidelity changing-reference composition

The single-fidelity changing-reference construction approximates a target-to-reference KR map by composing triangular maps trained in stages. Each new factor corrects the current map-induced approximation, with references updated between stages.

B.1 Single-fidelity changing-reference composition

  • Given target samples and a reference density, the goal is to approximate the KR map S satisfying S♯π = η.Source: §B.1 · p. 30
  • The construction trains stages from m = L down to m = 1, using the prescribed reference at the first stage and each prior stage’s map-induced approximation as the next reference.Source: §B.1 · pp. 30–31
  • The composed map uses successive factors, each correcting the discrepancy between the current map-induced approximation and the true target.This is the changing-reference composition described for deep transport maps.Source: §B.1 · p. 31

B.2 Single-fidelity fixed-reference composition

The fixed-reference construction progressively Gaussianizes target samples by composing triangular corrections while keeping the reference density unchanged. With a standard Gaussian reference, every stage retains the same log-concave, factorized-reference structure.

B.2 Single-fidelity fixed-reference composition

  • The cumulative map is updated by composition across stages, producing the final fixed-reference deep map.The construction trains from stage L down to stage 1.Source: §B.2 · p. 31
  • Each stage applies the current cumulative map to target samples, then learns a triangular correction that moves the transformed samples closer to the reference density.At the first stage, the cumulative map is the identity, so the transformed density is the target and the transformed samples are unchanged.Source: §B.2 · p. 31
  • Unlike the changing-reference construction, this strategy keeps the reference fixed; a standard Gaussian choice preserves the same log-concave, factorized-reference structure at every stage.Source: §B.2 · p. 31

B.3 Relation to the multifidelity hierarchy

The hierarchical multifidelity formulations extend the single-fidelity deep-map strategies by defining stages through prescribed fidelity levels. Their changing- and fixed-reference variants differ in how they use fidelity-specific samples.

B.3 Relation to the multifidelity hierarchy

  • The two hierarchical multifidelity formulations extend the single-fidelity deep-map strategies by incorporating the prescribed fidelity hierarchy into their stages.Source: §B.3 · p. 32
  • In the changing-reference method, the reference changes by stage and each stage uses samples from its prescribed fidelity rather than repeatedly refining one target distribution.Source: §B.3 · p. 32
  • In the fixed-reference method, the reference remains η, lower-fidelity samples are transformed by the lower-fidelity composition, and each new map corrects its Gaussianization.Source: §B.3 · p. 32

C Fixed-Reference Separability for Factorized References

For factorized references, fixed-reference objectives separate across triangular components when component parameters and regularization are separable. The appendix also specifies regularization selection and experimental map details.

C Fixed-Reference Separability for Factorized References

  • With a factorized reference, each fixed-reference hierarchical stage separates across triangular components, while each NH-MF component objective still couples high- and low-fidelity information.The separability follows from triangularity and a common factorized reference, not from one particular multifidelity construction.Source: C Fixed-Reference Separability for… · pp. 32–33

D Regularization selection

  • Regularization is selected by five-fold cross-validation per scenario, using reference-space validation and squared MMD averaged over four Gaussian-kernel bandwidths.The scenario-specific selected value is fixed for reported training replications; only the high-fidelity data are split into folds.Source: D Regularization selection · p. 34

E Additional details for Section 5.1

F Additional details for Section 5.2

G Additional details for Section 5.3

The appendix specifies the predator-prey simulation and inference setup, including how fidelity-specific training data are generated and how conditional transport samples are evaluated against SMC–ABC.

G Additional details for Section 5.3

  • The Section 5.3 setup uses a predator-prey process with Gillespie SSA at high fidelity, tau leaping at the first lower fidelity, and a mean-field ODE at the second.The SSA and tau-leaping/ODE configurations use their stated numerical settings; the reaction network is listed in Table G.1.Source: G Additional details for Section 5.3 · pp. 35–36, Table G.1 · p. 35
  • Training summaries comprise population means, log-variances, lag-1 and lag-2 autocorrelations, and cross-correlation, with independent Gaussian summary noise.Training datasets are independent subsamples from fidelity-specific pools, and map training does not use paired cross-fidelity data.Source: G Additional details for Section 5.3 · p. 36
  • For each prior-predictive observation, transport-generated conditional samples are compared in physical log-rate coordinates with a fixed-size SMC–ABC posterior subset.SMC–ABC uses the uniform physical prior, SSA simulations, prior-predictive normalization, and a Gaussian ABC kernel.Source: G Additional details for Section 5.3 · pp. 35, 37
  • The reported risk floors estimate metric values when map-generated samples are perfect draws from the SMC–ABC reference distribution.They are estimated from repeated pairs of independent sample subsets over the same prior-predictive observations.Source: G Additional details for Section 5.3 · pp. 37–38

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