The optimal transport (OT) map provides a geometric transformation for aligning probability distributions and has become a useful tool in machine learning. However, existing estimators of the OT map still exhibit a gap between sharp statistical guarantees and practical parametric estimation based on stable training objectives. Theoretical estimators achieve minimax optimal convergence rates, but they are typically nonparametric and can incur demanding implementation design or inference costs. Practical estimators are parametric and scalable, but their statistical guarantees remain underexplored, and their min-max, adversarial-like training objectives can be sensitive to optimization algorithms. We propose BROT (Barycentric Regression for OT), a simple two-step method that first computes the unregularized OT plan and then fits a deep neural network (DNN) to the induced barycentric targets by least-squares regression. Under standard regularity conditions, we prove that the DNN estimator of BROT attains the minimax convergence rate, when the ground-truth OT map is Lipschitz. Numerical studies on synthetic datasets and an image dataset show that BROT provides accurate map estimates, strong target distribution matching, and competitive transport costs, compared to existing estimation methods. Experiments on two downstream tasks, single-cell perturbation prediction and unsupervised domain adaptation, further suggest that the accurate estimation of BROT can translate into stronger task performance.
Figures & tables
Figure 1: Outline of our proposed framework BROT. We first compute the unregularized OT plan, and train a DNN to fit source data to barycentric targets obtained from the OT plan.
Figure 2: Statistical convergence to the ground-truth OT map. (Left) training data size n vs. estimation error. Points are the means across 10 random seeds and the shaded bands show the 10 -th to 90 -th percentile. (Right) theoretical rate n−1(logn)2 vs. estimation error for BROT, with a linear regression line and its relative RMSE (rel. RMSE).
Gaussian → Rectangle
Gaussian → Ellipse
AFHQv2 wild → cat
Method
Wass
TC
Wass
TC
Wass
TC
1NN Manole et al. (2024)
0.126
1095.67
0.207
1122.30
54.819
3296.54
LSOT Seguy et al. (2018)
0.111
1095.69
0.172
1124.35
53.263
2627.29
ICNN Makkuva et al. (2020)
0.211
1097.51
0.171
1122.54
∗
∗
OTP Choi et al. (2025b)
0.258
1099.17
0.596
1129.88
50.651
1408.23
DIOTM Choi et al. (2025a)
0.123
1098.21
0.228
1123.99
51.000
1453.99
Table 1: Comparison of map quality: distribution matching and transport cost. Best values of Wass are bolded and ‘ ∗ ’ marks the failed cases due to algorithm instability. Results are averaged over five random trials, and standard deviations are reported in Table 4 of Section B.3 .
Method
MMD ↓
L2 difference ↓
r -std ↑
TC ↓
Identity
0.028
1.213
0.917
0.000
1NN ( Manole et al., 2024 )
0.002
0.189
0.990
6.745
LSOT ( Seguy et al., 2018 )
0.012
0.287
0.982
4.381
ICNN ( Makkuva et al., 2020 )
0.006
0.258
0.987
4.329
OTP ( Choi et al., 2025b )
0.035
1.435
0.917
0.514
DIOTM ( Choi et al., 2025a )
0.002
0.255
0.990
3.993
Table 2: Comparison of single-cell perturbation prediction performance. The results are averaged across the 35 drug perturbations from 4i dataset Gut et al. (2018) . Among the methods that satisfy MMD≤0.03 (the MMD of Identity), the best value is bolded and the second-best is underlined for each metric. Standard deviations are reported in Table 6 of Section B.4 .
Method
USPS → MNIST
VisDA-17 (synthetic → real)
CLIP-ViT/B-32
DINOv2-ViT/B-14
CLIP-ViT/B-32
DINOv2-ViT/B-14
Identity
52.05
58.07
64.04
60.66
1NN Manole et al. (2024)
80.34
75.25
74.17
74.39
LSOT Seguy et al. (2018)
30.12
52.20
76.87
81.57
ICNN Makkuva et al. (2020)
75.76
54.70
74.12
77.77
OTP Choi et al. (2025b)
49.98
57.36
76.15
64.51
Table 3: Comparison of unsupervised domain adaptation performance. The performance is evaluated by the per-class mean accuracy (%) on the target-domain test data. Best results are bolded and the second places are underlined , and the standard deviations are in Table 7 of Section B.5 .
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3: Source P (left) and target Q=(T0)#P (right) used in Section 5.1 . The color of each point indicates the radius of data ∥x∥.
Figure 4: Source (purple) and target (red) data for the two synthetic cases used in Section 5.2 .
Source → Target
Method
Wass ↓
TC ↓
Gaussian → Rectangle
1NN
0.126±0.008
1095.67±0.61
LSOT
0.111±0.003
1095.69±0.14
ICNN
0.211±0.005
1097.51±0.09
OTP
0.258±0.008
1099.17±2.39
DIOTM
0.123±0.007
1098.21±2.35
BROT
0.110±0.003
1095.65±0.05
Appendix
Table 4: Comparison of map quality: distribution matching and transport cost. Results are reported as mean ± standard deviation across five random trials. Best values of Wass are bolded .
Method
Synthetic cases
AFHQv2 wild → cat
Train (%)
Inference (%)
Train (%)
Inference (%)
1NN [ 36 ]
20%
339%
17%
486%
LSOT [ 48 ]
267%
95%
154%
108%
ICNN [ 35 ]
2129%
87%
4985%
229%
OTP [ 8 ]
1078%
105%
785%
286%
DIOTM [ 7 ]
1445%
81%
2958%
205%
Appendix
Table 5: Comparison of computation time. Relative training and inference computation times (BROT =100% ). We report mean relative training and inference computation times (BROT =100% ) over five random seeds, and synthetic cases are the average over the two datasets.
Method
MMD ↓
L2 difference ↓
r -std ↑
TC ↓
Identity
0.028 ± 0.029
1.213 ± 0.899
0.917 ± 0.081
0.000 ± 0.000
1NN [ 36 ]
0.002 ± 0.000
0.189 ± 0.062
0.990 ± 0.009
6.745 ± 6.108
LSOT [ 48 ]
0.012 ± 0.002
0.287 ± 0.121
0.982 ± 0.013
4.381 ± 5.437
ICNN [ 35 ]
0.006 ± 0.001
0.258 ± 0.096
0.987 ± 0.008
4.329 ± 5.751
OTP [ 8 ]
0.035 ± 0.029
1.435 ± 0.852
0.917 ± 0.081
0.514 ± 0.141
DIOTM [ 7 ]
0.002 ± 0.001
0.255 ± 0.101
0.990 ± 0.010
3.993 ± 5.875
Appendix
Table 6: Comparison of single-cell perturbation prediction performance. The results are reported as mean ± standard deviation across the 35 drug perturbations. Among the methods that satisfy MMD≤0.03 , the best value is bolded and the second-best is underlined for each metric.
USPS → MNIST
VisDA-17 (synthetic → real)
Method
CLIP-ViT/B-32
DINOv2-ViT/B-14
CLIP-ViT/B-32
DINOv2-ViT/B-14
1NN [ 36 ]
80.34 ± 0.34
75.25 ± 0.77
74.17 ± 0.08
74.39 ± 0.01
LSOT [ 48 ]
30.12 ± 2.84
52.20 ± 0.88
76.87 ± 0.63
81.57 ± 0.18
ICNN [ 35 ]
75.76 ± 4.63
54.70 ± 21.08
74.12 ± 1.07
77.77 ± 0.41
OTP [ 8 ]
49.98 ± 4.79
57.36 ± 5.18
76.15 ± 0.42
64.51 ± 0.37
DIOTM [ 7 ]
82.71 ± 1.00
77.46 ± 0.51
76.45 ± 0.55
27.05 ± 35.31
Appendix
Table 7: Comparison of unsupervised domain adaptation performance. Per-class mean accuracy (%) on the target-domain test data with standard deviations across three random seeds, reported as mean ± standard deviation.
Method
Train
Backbone
Accuracy
Encoder
Head
DeepJDOT [ 13 ]
✓
✓
ResNet-50
68.00
JUMBOT [ 17 ]
✓
✓
ResNet-50
72.50
ELOT [ 53 ]
✓
✓
ResNet-50
76.32
✗
✓
CLIP-ViT/B-32
79.34
✗
✓
DINOv2-ViT/B-14
81.59
Appendix
Table 8: Comparison of domain adaptation performance for BROT and end-to-end OT-based domain adaptation methods. The performance is evaluated by the per-class mean accuracy (%) on the target-domain test data, following [ 40 ] . The results of BROT are averaged over three random seeds, while baseline results are copied from the original papers.
Figure 5: Sensitivity of DNN architecture and Jacobian penalty to the statistical convergence to the ground-truth OT map. (Left) Three MLP architectures (Jacobian penalty weight fixed to 10−2 ). (Right) Three Jacobian penalty weights (architecture fixed).
Many modern generative modeling methods, including diffusion models, normalizing flows, and flow matching, estimate transport maps or plans between distributions without explicitly targeting an optimal transport (OT) map. In applications like generative modeling, the transport cost itself is irrelevant, and this makes it natural to target maps which are more tractable from either a statistical or computational standpoint. In this short note, we formalize the task of estimating any valid transport map in a rigorous minimax framework. One consequence of this framing is that it yields sample complexity lower bounds for any method whose learned object is evaluated as a transport map or plan, including flow matching and diffusion-based generative models, in settings where direct analysis would be challenging due to the analytic complexity of the methods and their target maps. We observe that, under standard, though strong, stability assumptions from the OT literature, estimating any valid transport map is statistically as hard as estimating the OT map. We complement these results with some examples showing that when these stability assumptions fail, alternative transport maps can be learned substantially more accurately than the OT map. Our minimax framing provides a rigorous foundation for understanding the statistical limits of modern transport-based generative methods and clarifies when targeting sub-optimal maps can provide real statistical advantages.
Sivaraman Balakrishnan
Department of Statistics and Data Science† · Machine Learning Department‡
Optimal Transport (OT) provides a principled framework for learning transformations between probability distributions from unpaired samples. In many applications, however, a single transformation must map several source distributions to a common target distribution. For example, image restoration might require handling different types of degradation without knowing the degradation of each input at inference time. Simple approaches of pooling the source distributions only encourage alignment with the target at the aggregate level and may leave individual sources misaligned. In our paper, we consider the simultaneous OT problem which formalizes the task of learning a shared transport map that minimizes the average transport cost while aligning each source distribution with a prescribed target. We propose a neural method for solving the simultaneous OT problem by learning a shared transport map that minimizes the average transport cost while aligning each source distribution with a prescribed target. We derive a max-min formulation for learning this map. We illustrate its application to image restoration, where a single model handles multiple degradation types using a common collection of clean target images.
Milena Gazdieva, Kirill Sokolov, Jiawei Chen +2
Applied AI Institute AXXX Moscow, Russia · Lomonosov Moscow State University Moscow, Russia · MIRIAI Moscow, Russia
We propose a new regularized optimal transport (OT) formulation, termed sliced-regularized optimal transport (SROT). Unlike entropic OT (EOT), which regularizes the transport plan toward an independent coupling, SROT regularizes it toward a smoothened sliced OT (SOT) plan. To the best of our knowledge, SROT is the first approach to leverage a version of SOT plan as a reference to improve classical OT. We provide a formal definition of SROT, derive its dual formulation, and provide a post-Bayesian interpretation of SROT. We then develop a Sinkhorn-style algorithm for efficient computation, retaining the same scalability advantages as EOT. By incorporating a scalable SOT plan as a prior, SROT yields more accurate approximations of the exact OT plan than EOT under the same level of regularization. Moreover, the resulting transport plan improves upon the reference SOT plan itself. We further introduce the corresponding OT divergence induced by SROT, named SROT divergence, and analyze its topological and computational properties. Finally, we validate our approach through experiments on synthetic datasets and color transfer tasks, demonstrating that SROT is better than both EOT and SOT in approximating exact OT. Additional experiments on gradient flows further highlight the advantages of SROT divergence.