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Towards Universal Wasserstein Barycenters through Flow Matching

Eduardo Fernandes Montesuma

arXiv:2609.38547v1cs.LGcs.AIstat.ML

TL;DR

Approximating Wasserstein barycenters for every weight vector remains underexplored, particularly for joint feature-label measures. BaryFM uses dual optimal transport and flow matching to sample from this family. It achieves the best average rank among 15 tested methods over 10 domain-adaptation benchmarks.

  • Problem

    Approximating the full family of Wasserstein barycenters over the simplex is underexplored, and existing neural or universal methods do not handle joint feature-label measures.

    Source: §1 · pp. 1–2

  • Method

    BaryFM combines a dual-OT head with a flow-matching head to transport marginals into barycenters across the weight simplex, including for joint measures.

    Source: §5 · p. 9

  • Results

    BaryFM has the best average rank among 15 tested methods over 10 domain-adaptation benchmarks.

    Source: §5 · p. 9

  • Takeaways & Limitations

    The evaluation extends barycenter solvers to domain generalization, where barycenter augmentation improves over ERM and is better than or competitive with Domain MixUp.

    Source: §5 · p. 9

    BaryFM computes a regularized barycenter that has inherent biases relative to the true Wasserstein barycenter, and further analysis of this difference is needed.

    Source: §5 · p. 9

Abstract

from arXiv · show

Defining a weighted mean over probability measures under probability metrics is a central tool in probabilistic machine learning. Under the Wasserstein metric, these are called \emph{Wasserstein barycenters}. While most approaches compute barycenters for a fixed weight vector, approximating the whole family of barycenters over the simplex, which we call the \emph{Wasserstein simplex}, remains underexplored. We refer to this problem as \emph{Universal Barycenter Approximation}, and propose \texttt{BaryFM}, a flow matching model transporting the marginal measures into any barycenter in the Wasserstein simplex. Once trained, the network can draw samples from measures in the Wasserstein simplex through an ordinary differential equation. We validate our method on 4 downstream tasks: domain adaptation, generalization, Bayesian posterior aggregation and algorithmic fairness. \texttt{BaryFM} achieves the best average rank among 15 competing methods across 10 domain adaptation benchmarks, matching or surpassing non-universal solvers.

1 INTRODUCTION

The paper studies the underexplored problem of approximating the full family of Wasserstein barycenters over the simplex, rather than a barycenter for one fixed weight vector. It proposes BaryFM as a universal neural solver and evaluates it across four downstream applications.

1 INTRODUCTION

  • The Wasserstein simplex is the family of barycenters over all weight vectors, a problem left open beyond restricted cases such as Gaussian measures (Fan et al., 2020, Appendix B).The paper frames approximation of this whole family as Universal Barycenter Approximation.Source: §1 · p. 1
  • Most methods amortize sampling for a fixed barycenter, but generally require a new optimization for each queried weight vector.Fan et al. (2020, Appendix B) is identified as an exception, validated in restricted scenarios such as Gaussian measures.Source: §1 · pp. 1–2
  • Domain adaptation and generalization require barycenters over joint feature-label measures, which existing neural or universal methods do not handle.The relevant space is described as features paired with soft labels.Source: §1 · p. 2
  • BaryFM is proposed as the first neural universal barycenter solver that handles joint measures, using a shared encoder with velocity-field and dual-OT heads.The model transports marginal measures to any barycenter in their Wasserstein simplex.Source: §1 · p. 2
  • The evaluation covers four downstream applications: domain adaptation, generalization, algorithmic fairness, and Bayesian posterior aggregation.The paper compares BaryFM with prior art across these applications.Source: §1 · p. 2

2 BACKGROUND

The background introduces Wasserstein barycenters as minimizers of a weighted transport objective and describes conditional flow matching as a way to learn flows between measures. It also gives the uniqueness assumption used for the paper’s barycenter definition.

2.1 OPTIMAL TRANSPORT AND BARYCENTERS

  • Optimal transport can be formulated through maps or couplings, with entropic regularization yielding an approximate transport problem.The background presents primal and dual formulations and defines Wasserstein distance as the unregularized case.Source: §2.1 · pp. 2–3
  • The barycenter problem seeks a measure minimizing the weighted transport costs to the input marginals for coordinates in the simplex.The paper assumes all marginals vanish on small sets to ensure a unique solution for every weight vector.Source: §2.1 · p. 3

2.2 CONDITIONAL FLOW MATCHING

  • Conditional flow matching learns a time-dependent velocity field that transports a source measure into a target measure.Its conditional velocity target is an expectation that is intractable because it requires the posterior over endpoint pairs given an interpolated point.Source: §2.2 · pp. 3–4
  • Using an optimal-transport coupling to sample endpoint pairs extends conditional flow matching to flows between arbitrary source and target measures.With an independent product coupling, the objective reduces to the earlier conditional flow matching formulation.Source: §2.2 · p. 4

3 BA R YFM: AMORTIZED, UNIVERSAL WASSERSTEIN BARYCENTERS

BaryFM jointly learns transport-related quantities and a conditional velocity field so it can approximate and sample barycenters across the simplex. Its design also extends the flow to feature-label joint measures while keeping labels in the simplex.

3 BA R YFM: AMORTIZED, UNIVERSAL WASSERSTEIN BARYCENTERS

  • A shared encoder conditions the velocity and dual-OT heads on the sample, source index k, and weight vector λ.The dual-OT head estimates OT quantities, while the velocity head transports samples from a marginal toward the estimated barycenter.Source: Figure 2 · p. 4, §3 · pp. 4–5
  • The dual-OT head approximates transport between each marginal and the moving barycenter target, avoiding K separate OT solves when estimating that target.The method uses a neural barycentric map as an approximation to the Monge map when the latter exists, as entropic regularization tends to zero.Source: §3 · p. 5
  • For flow matching, BaryFM draws source-target pairs from an estimated conditional OT plan and trains on their linear interpolants.The source sample is drawn from p_k, and the target is sampled conditionally from the estimated plan.Source: §3 · p. 5
  • Training samples λ across the simplex and alternates dual-OT ascent with velocity-loss minimization, so both heads are trained for varying barycentric coordinates.For fixed dual parameters, the velocity update reduces to conventional OT-CFM.Source: §3 · p. 5
  • After training, one ODE solve transports each marginal sample to a barycenter for the queried λ, amortizing sampling over both particles and barycentric coordinates.For feature-label joint measures, the ODE step projects labels back onto the label simplex.Source: §3 · pp. 6–7

4 EXPERIMENTS

The experiments evaluate Wasserstein barycenter solvers on domain adaptation and domain generalization. BaryFM ranks best across the reported domain-adaptation benchmarks and BaryFMRandom ranks best among the tested generalization augmentations.

4.1 DOMAIN ADAPTATION

  • The domain-adaptation task tests multi-source alignment to an unlabeled target through an intermediate barycentric domain, while also rewarding preservation of class structure.The evaluation uses the Wasserstein Barycenter Transport pipeline.Source: §4.1 · p. 7
  • BaryFM achieves the best average rank over the tested domain-adaptation benchmarks, with top-1 performance on 6 of 10 benchmarks.WGF is the next-best method in average rank; the comparison includes 15 methods.Source: §4.1 · p. 7, Figure 7 · p. 8

4.2 DOMAIN GENERALIZATION

  • BaryFM’s domain-generalization augmentation blends source domains nonlinearly through Wasserstein barycenters, unlike the linear mixing used by MixUp variants.The compared strategies include MixUp, Domain MixUp, BaryFMRandom, and BaryFMVertex; the latter samples simplex vertices.Source: §4.2 · p. 8
  • On the reported generalization benchmarks, augmentations broadly help, Wasserstein interpolation performs better than MixUp variants, and BaryFMRandom works better overall than BaryFMVertex.The strategy encodes source-domain data into a shared latent space, generates synthetic simplex samples, and decodes them; the experiments use VAE latents for images and PCA for EEG.Source: §4.2 · pp. 8–9, Figure 8 · p. 9
  • BaryFMRandom ranks best on average (1.40), ahead of BaryFMVertex (2.80), Domain MixUp (3.00), MixUp (3.20), and ERM (4.60).Table 1 reports downstream performance of ResNets and CBraMod on five multi-domain benchmarks.Source: §4.2 · p. 8, Table 1 · p. 9

5 CONCLUSION

The paper introduces BaryFM for approximating Wasserstein barycenters across the simplex, including feature-label joint measures. It reports strong results on domain adaptation and generalization, while noting that its regularized barycenters can differ from the true Wasserstein barycenter.

5 CONCLUSION

  • BaryFM is a neural universal barycenter method with a dual OT head and a flow-matching head, and it handles flows over feature-label joint measures.The dual head approximates OT between marginals and barycenter targets; the flow-matching head models transport between them.Source: §5 · p. 9
  • BaryFM has the best average rank among 15 tested methods over 10 domain-adaptation benchmarks, and Wasserstein-barycenter augmentation improves over ERM and is better or competitive against Domain MixUp.The paper validates BaryFM on domain adaptation, generalization, algorithmic fairness, and Bayesian posterior aggregation.Source: §5 · p. 9
  • BaryFM computes a regularized barycenter with inherent biases relative to the true Wasserstein barycenter, and further analysis of this difference remains future work.In restricted cases, including Location-Scatter families, the method is reported to remain accurate to a known ground truth despite these biases.Source: §5 · p. 9

Supplementary Materials for Towards Universal Wasserstein Barycenters through

The supplement outlines implementation and experimental details, then reviews barycenter solvers and their limitations. It emphasizes that approximating barycenters across the Wasserstein simplex remains largely unaddressed by prior methods.

A INTRODUCTION

  • The supplement covers alternative BaryFM variants, toy and downstream experiments, and additional implementation, hyperparameter, and experimental details.Its experiment appendices include Bayesian posterior aggregation, fairness, domain adaptation, and domain generalization.Source: A INTRODUCTION · p. 17

B RELATED WORK

  • Empirical solvers can require re-solving to draw new samples and may require complete marginal samples, while parametric Gaussian and GMM solvers rely on assumptions that may not fit real data.Gaussian fixed-point complexity scales as O(d^3); diagonal-covariance GMM methods scale with sample count and dimensionality.Source: B RELATED WORK · p. 17
  • Neural barycenter methods include multi-level optimization, direct neural parametrizations of the barycenter, and methods that model dynamics toward the barycenter.Examples include ICNN approaches, latent-to-barycenter methods, and flow or bridge-based approaches.Source: B RELATED WORK · p. 17
  • Most prior methods require a new optimization for each weight vector; the prior universal solver cited here was validated only on low-dimensional toy examples.The supplement’s comparison taxonomy includes parametrization, optimization, amortization, universality, and handling of joint measures.Source: B RELATED WORK · p. 17, Table 2 · p. 18
  • The supplement categorizes solver biases as input, algorithmic, and output biases, including mini-batch or parametric approximations, modeling choices, and ODE time discretization.The taxonomy is presented as a set of design choices that can affect the computed barycenter.Source: B RELATED WORK · p. 18, Table 3 · p. 18

C ALTERNATIVE VERSIONS OF BARYFM

The appendix compares alternative BaryFM formulations and explains the roles of its components. It also describes empirical and neural fixed-point variants, a semi-unbalanced formulation, and an alternative ODE integration strategy.

C ALTERNATIVE VERSIONS OF BARYFM

  • The alternative-method ladder includes empirical BaryFM, NeuralFP without flow matching, and semi-unbalanced BaryFM; the dual OT and velocity heads amortize OT estimation and sampling, respectively.The dual head replaces repeated mini-batch OT problems with a neural-network forward pass, while the velocity head samples for arbitrary λ by solving an ODE.Source: C ALTERNATIVE VERSIONS OF BARYFM · p. 19, Table 4 · p. 19

C.1 EMPIRICAL BARYFM

  • Empirical BaryFM updates barycenter support through fixed-point iterations using empirical OT and a barycentric mapping, but solves K OT problems within each iteration.Its stated complexity reduces from O(LFP LSinkhorn K B^2) for the empirical solver to O(LFP K B^2) for the neural solver, which estimates OT variables in one pass.Source: §C.1 · pp. 19–20

C.2 NEURAL FIXED-POINT

  • NeuralFP removes the velocity field and uses a learned dual-OT network in a fixed-point strategy; its parametrization is independent of barycenter support and uses K neural networks.Inference replaces the fixed-point barycentric mapping with the neural mapping; the comparison notes WIN uses 2K + 1 networks.Source: §C.2 · pp. 20–21

C.3 SEMI-UNBALANCED BARYFM

  • The semi-unbalanced formulation relaxes the marginal constraint on source measures while retaining the barycenter-side mass constraint to avoid degenerate null-mass barycenters.The first-marginal violation is penalized with weight τ.Source: §C.3 · p. 22

C.4 OTHER ODE INTEGRATION STRATEGIES

  • The appendix compares Euler and midpoint ODE integration strategies and notes that midpoint generally has smaller discretization error for the same time step.The comparison is shown in Figure 12.Source: §C.4 · p. 22

D SWISS ROLL MEASURES

The Swiss-roll experiment uses location-scatter measures whose barycenters are known analytically, enabling comparison across the whole simplex; BaryFM fits the manifold well and is substantially faster than WGF.

D SWISS ROLL MEASURES

  • The experiment uses affine location-scatter transforms of a centered identity-covariance Swiss-roll reference, giving analytically known barycenters and a ground truth for scoring.The illustrated simplex contains three marginals.Source: D SWISS ROLL MEASURES · p. 23, Figure 9 · p. 23
  • Across the simplex, BaryFM fits the Swiss-roll manifold well, while WGF and BaryFM are more accurate than NWBU; WGF’s support is unusually tight under entropic regularization.NWBU barycenters are qualitatively correct but considerably noisier.Source: D SWISS ROLL MEASURES · p. 24, Figure 11 · p. 24
  • BaryFM is around 4 orders of magnitude faster than WGF while achieving parity with WGF in BW2-UVP%.Its sampling speed sits between NWBU and WGF as velocity-field evaluations and seconds per barycenter vary.Source: D SWISS ROLL MEASURES · p. 24, Figure 12 · p. 24

E BAYESIAN POSTERIOR AGGREGATION

The study evaluates Wasserstein barycenters as posterior aggregates against a full-data posterior, then examines both solver performance and the broader simplex of candidate barycenters. BaryFM-U with k-CFG is statistically tied with U-NOT in the homogeneous setting and outperforms it in the heterogeneous setting, while simplex exploration can improve posterior similarity beyond uniform weights.

Dataset and evaluation

  • The evaluation compares barycentric measures with the full-data posterior using BW2-UVP% and 2-Sliced-Wasserstein, based on three partitions of Wine logistic-regression data.The full-data posterior is treated as ground truth, and the reported samples come from three partition posteriors and that full posterior.Source: E BAYESIAN POSTERIOR AGGREGATION · p. 25

Main Results

  • With k-CFG, BaryFM-U reaches 8.02±0.07 versus U-NOT’s 8.08±0.10 on the homogeneous benchmark, and 9.93±0.17 versus 10.26±0.08 on the heterogeneous benchmark.The authors describe the homogeneous results as statistically tied and the heterogeneous result as surpassing U-NOT.Source: E BAYESIAN POSTERIOR AGGREGATION · p. 25, Table 5 · p. 26

On the benefits of universality

  • Universality improves posterior-aggregation ranks in most reported comparisons, but BaryFM-U’s heterogeneous rank worsens from 2nd to 6th.Ranks improve from 11th to 7th and 10th to 7th in the homogeneous and heterogeneous settings, respectively; semi-unbalanced homogeneous rank improves from 10th to 5th.Source: E BAYESIAN POSTERIOR AGGREGATION · p. 26

Classifier Free Guidance

Hyperparameters and their sensitivity

  • Sensitivity is more pronounced for entropic regularization, which generally favors ε ≤ 2; performance varies less with batch size and fixed-point iterations than with seed standard deviation.The study tests multiple values of ε, batch size B, and fixed-point iteration count LFP.Source: E BAYESIAN POSTERIOR AGGREGATION · pp. 27–28, Figure 15 · p. 28

Conclusion

  • Exploring the simplex improves heterogeneous balanced BaryFM’s BW2-UVP% against the true posterior from 10.51 to 8.91; the best reported values are 7.69 in the homogeneous and 8.79 in the heterogeneous setting.The best posterior approximation is not necessarily at the uniform barycenter.Source: E BAYESIAN POSTERIOR AGGREGATION · pp. 26, 28, Figure 13 · p. 26

F FAIRNESS

The fairness experiments assess the tradeoff between disparate impact and classification accuracy, including how both change during transport to a barycenter. The supplied domain-adaptation passages describe barycenter transport and its scalability constraint, but do not report its benchmark outcome.

F FAIRNESS

  • On Adult, BaryFM repair raises DI from 0.509 to 1.013 at a 1.16% classification-accuracy cost; empirical solvers tend to favor fairness, while neural solvers tend to preserve accuracy with residual fairness gaps.Across the reported fairness results, most solvers improve fairness; the comparison spans four datasets.Source: F FAIRNESS · pp. 29–30, Table 8 · p. 30
  • As flow time advances, DI improves toward 1.0 while downstream classification accuracy degrades, tracing the fairness–task-performance tradeoff during transport.At t = 0, the trajectory starts at the un-repaired baseline.Source: F FAIRNESS · p. 30, Figure 17 · p. 31
  • The universal solver with LFP = 10 has geometric-mean fairness rank 1.86 versus DWB’s 2.21, and accuracy rank 3.94 versus U-NOT’s 3.87 and NWBU’s 3.53.The reported Univ−5 variant balances disparate impact and classification accuracy; the figure compares average method ranks across datasets.Source: Figure 16 · p. 30, F FAIRNESS · pp. 31–32, Figure 18 · p. 31

G DOMAIN ADAPTATION

  • The domain-adaptation evaluation uses a multi-source-to-target setting with distribution shift and tests barycenter solvers on alignment of heterogeneous domains and preservation of class structure.The described pipeline uses source-domain embeddings and an intermediate barycenter.Source: G DOMAIN ADAPTATION · p. 32, §G.1 · p. 32

G.1 WASSERSTEIN BARYCENTER TRANSPORT

  • Wasserstein Barycenter Transport first aggregates source domains into a barycenter, then transports it to the target; the stated scalability boundary is large OT problems such as DomainNet’s roughly 170,000-point domains.The authors address this constraint with a CFM alignment problem and transport samples through an ODE.Source: §G.1 · p. 32

G.2 EXPERIMENTAL DETAILS

Across 10 benchmarks, BaryFM achieves an average rank of 1.75, surpassing WGF; the experiments also examine its ablations, inference sensitivity, and parameter scaling. Benchmark-specific results range from clear BaryFM wins to settings where gains are marginal or WGF matches it.

G.2 EXPERIMENTAL DETAILS

  • 1.75 average rank across 10 benchmarks puts BaryFM ahead of WGF, the previous state-of-the-art barycenter solver for domain adaptation.Results aggregate average target-domain performance for vision and TEP benchmarks and pooled target-subject performance for neuroscience datasets.Source: §G.2 · p. 34, Table 13 · p. 35
  • Inference sensitivity varies by dataset: Office 31 benefits most from more generated points, TEP is stable across their number, and performance is generally robust to ODE settings.The reported parameter counts remain bounded as marginal measures increase for BaryFM architectures; the discussion identifies no consensus on neural-solver architecture design.Source: §G.2 · pp. 35–36, Figure 19 · p. 36

G.3.1 OFFICE 31

  • BaryFM variants rank first or second on Office 31, and BaryFM alone improves over source-only on Amazon as well as dSLR and Webcam.NWB-U falls below source-only on dSLR: 91.37±2.25 versus 99.41±0.51.Source: §G.3.1 · p. 37, Table 16 · p. 37

G.3.2 OFFICE HOME

  • BaryFM variants achieve the best Office Home performance, with BaryFM at 78.72±0.12 versus WGF at 78.03±0.13.Only WGF variants and BaryFM improve over the source-only baseline.Source: §G.3.2 · p. 38, Table 17 · p. 38

G.3.3 DOMAIN NET

  • DomainNet gains are marginal: TSBM ranks first at 52.59±0.02, while BaryFM and TDSB each reach 52.56 with their reported standard deviations.BaryFMunsup reaches 52.70±0.04 among BaryFM variants; WGF is the only empirical solver reported to improve over source-only.Source: §G.3.3 · p. 39, Table 18 · p. 39

G.3.4 BCI-CIV-2A

  • All methods improve over baseline on BCI-CIV-2a, and BaryFM methods achieve the best results, with BaryFMunsup performing best.The comparison reports pooled accuracy on test subjects.Source: §G.3.4 · p. 40, Table 19 · p. 40

G.3.5–G.3.9 NEUROSCIENCE BENCHMARKS

  • In a benchmark, BaryFM records 57.42±0.94 versus second-best TSBM’s 55.03±0.35; in another, it achieves 63.70±0.15, statistically indistinguishable from WGF at 63.70±0.11.Source: §G.3.6 · p. 42, §G.3.9 · p. 45

G.3.10 TENNESSEE EASTMAN PROCESS

  • BaryFMunsup performs best on TEP at 87.32±1.15 versus WGF’s 86.87±1.37, and all methods significantly improve over source-only.Source: §G.3.10 · p. 46, Table 26 · p. 46

H.1 DATA AUGMENTATION FOR DOMAIN GENERALIZATION

For domain generalization, BaryFM augments source data with samples from barycenters over the Wasserstein simplex. The comparison with linear mixing reports different interpolation artifacts and broader latent-space coverage for BaryFM.

H.1 DATA AUGMENTATION FOR DOMAIN GENERALIZATION

  • BaryFMRandom augments training data with samples from barycenters at weights drawn from a Dirichlet distribution, then trains a model on the augmented pool.The alternative pool uses simplex vertices, roughly corresponding to sampling from marginal measures.Source: §H.1 · p. 47
  • Linear Domain MixUp produces ghosting artifacts, while BaryFM interpolation produces fuzziness; both generate samples spanning new latent-space regions.The passages describe image mixing in VAE latent space and EEG mixing in PCA space.Source: §H.1 · pp. 47–48, Figure 22 · p. 47, Figure 23 · p. 48
  • For uniform weights, BaryFM concentrates less than linear mixing, and its broader latent-space coverage is offered as an explanation for its performance gains over MixUp and Domain MixUp on Office Home.Source: §H.1 · p. 48

H.2 EXPERIMENTAL DETAILS

The experimental details describe latent-space preprocessing and the BaryFM architecture used for domain generalization. PCA dimensionality is selected using reconstruction and downstream metrics, with task performance matching raw signals around 2048 components.

H.2 EXPERIMENTAL DETAILS

  • Image experiments encode resized images into VAE latent codes of shape 4 × 32 × 32 before training BaryFM on those codes.The VAE downsamples images by a factor of 8; Figure 24 shows original and reconstructed images.Source: §H.2 · pp. 48–49, Figure 24 · p. 48
  • The domain-generalization BaryFM uses a U-Net for the feature-velocity component and its pooled bottleneck for the label-velocity component, with conditioning embeddings modulating residual blocks.The architecture also includes spatial self-attention and a dual OT head.Source: §H.2 · p. 49, Figure 25 · p. 49, Table 27 · p. 49
  • Around 2048 principal components, downstream performance matches raw EEG signals, so the experiments use 2048 dimensions for BCI-CIV-2a and 4096 for SEED-VIG.Figure 26 tracks channel-wise BW2-UVP%, log-spectral distance, MSE, and downstream performance as component count changes.Source: §H.2 · p. 50, Figure 26 · p. 50

H.3 DETAILED RESULTS

Across the reported generalization benchmarks, BaryFMRandom is the best augmentation method on three of five datasets and is more consistent than BaryFMVertex. Results vary by dataset: BaryFMRandom leads on Office 31, Office Home, and SEED-VIG, while Domain MixUp leads on DomainNet and BaryFMVertex leads on BCI-CIV-2a.

H.3 DETAILED RESULTS

  • Domain generalization results can differ from domain adaptation because generalization fine-tunes the full network from synthetic data and has no target-domain data.The domain adaptation pipeline runs over embeddings and can be viewed as linear probing.Source: §H.3 · p. 51
  • 85.94±0.66 vs. Domain MixUp’s 84.97±0.69: BaryFMRandom achieves the best rank on Office 31.The reported gap is 0.97, which the authors say exceeds seed noise.Source: §H.3 · p. 51, Table 29 · p. 51
  • BaryFMRandom is the best method on Office Home; all methods except Domain MixUp improve on ERM beyond the seed standard deviation.BaryFM samples become more blurred and distributed in latent space, while Domain MixUp samples show some ghosting and concentrate as weights move away from simplex vertices.Source: §H.3 · p. 51, Table 30 · p. 51
  • Domain MixUp is best and BaryFMRandom second on DomainNet, where both improve over ERM despite smaller overall gains.The benchmark has 345 classes and severe domain shift; the largest reported difference is Domain MixUp versus ERM at δ = 1.06.Source: Table 31 · p. 51, §H.3 · p. 52
  • BaryFMVertex improves over ERM by 3.54% on BCI-CIV-2a, while BaryFMRandom improves by 3.09.The evaluation uses subjects 1–5 as sources, 6–7 for checkpoint selection, and 8–9 for testing; EEG results have wider confidence intervals than image benchmarks.Source: §H.3 · p. 52, Table 32 · p. 52
  • BaryFMRandom improves Pearson correlation by 3.78% on SEED-VIG, above the MixUp and Domain MixUp gains of 2.81 and 2.74.Subjects 1–13 are sources, 14–17 are test subjects, and the remaining subjects are used for checkpoint selection; this split differs from domain adaptation.Source: §H.3 · p. 52, Table 33 · p. 52
  • BaryFMRandom is the best augmentation method on 3 of 5 datasets and is more consistent than BaryFMVertex.The authors attribute this result to an advantage from mixing domain characteristics.Source: §H.3 · p. 52
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