CyclOT: Learning Quadratic Optimal Transport Maps via Synchronized Forward-Backward Interpolants

Shizhou Xu; Jiachen Liu; Shih-Hsin Wang; Stefan Broecker; Yuhao Huang; Bao Wang; Thomas Strohmer

math.OCSep 12, 2026

CyclOT: Learning Quadratic Optimal Transport Maps via Synchronized Forward-Backward Interpolants

Authors: Shizhou Xu, Jiachen Liu, Shih-Hsin Wang, Stefan Broecker, Yuhao Huang, Bao Wang, Thomas Strohmer

Organizations: SLAC National Accelerator Laboratory, Stanford University, Menlo Park, CA 94025, USA · Department of Mathematics, University of California, Davis, Davis, CA 95616, USA · Department of Computer Science and Information Engineering, National Taiwan University, Taipei 10617, Taiwan · Department of Mathematics and Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84112, USA ·

Abstract

We study the recovery of forward and reverse quadratic optimal-transport maps from unpaired samples in high dimensions. We introduce a bidirectional neural framework in which the learned maps induce forward and backward displacement interpolants, while the training objective combines bidirectional quadratic action, discriminator-restricted Jensen-Shannon endpoint objectives, and two-sided cycle consistency. The construction requires neither precomputed sample pairings nor an explicit convex-potential parameterization. For absolutely continuous probability measures supported on a compact convex set, and under the stated generator-approximation, discriminator-richness, and minimizer-attainment conditions, we prove a population recovery theorem: for every prescribed accuracy, the sum of the corresponding $L^2$ errors between any global minimizer and the forward and reverse quadratic Brenier maps is below that accuracy, provided the discriminator level is sufficiently large and the annealing action weight becomes sufficiently small. Moreover, the cycle loss is bounded above by $λW_2^2(μ_0,μ_1)$ . Complementary results quantify approximate invertibility and show that exact endpoint Jensen-Shannon divergence and cycle consistency control missing target mass and many-to-one map collapse, respectively. Experiments on Swiss roll, MNIST, CelebA, single-cell perturbation data, and chest X-ray images evaluate endpoint fidelity, transport cost, inverse consistency, and the geometry of the induced interpolations.

Figures & tables

Panel of Figure 1 — Figure 1: Overview of our proposed target-free OT learning via synchronized forward–backward interpolants. The forward map $f_{\theta}$ transports $\mu_{0}$ from left to right, while the backward map $g_{\phi}$ transports $\mu_{1}$ from right to left; Both $f_{\theta}$ and $g_{\phi}$ are neural network-approximated maps. The forward and backward induced marginals are matched at the two endpoints, providing marginal supervision for the transport maps.

\{x_{0}^{i}\}_{i=1}^{B}\sim\mu_{0},\qquad\{x_{1}^{i}\}_{i=1}^{B}\sim\mu_{1}.

Algorithm 1 Forward-Backward Neural Optimal Transport

Panel of Figure 2 — Figure 2: Qualitative visualization of the Gaussian-to-Swiss-roll experiment. The left panel shows pointwise map-induced interpolation paths from source samples to transported endpoints. The right panel shows forward marginal matching, reverse marginal matching, and target-domain cycle reconstruction.

Method	Global map?	Cost $\downarrow$	TDR $\uparrow$	TPS $\uparrow$	FID $\downarrow$	Class-TV $\downarrow$	Train Time (min) $\downarrow$
Sinkhorn	No	245.201	0.992	0.984	93.773	0.072	–
CyclOT	Yes	212.43	0.957	0.954	92.269	0.039	7
Neural OT	Yes	133.68	0.956	0.952	120.45	0.130	198
ICNN OT	Yes	210.82	0.805	0.749	1565.65	0.162	43
Rectified flow	Yes	246.18	0.565	0.567	1282.97	0.171	29
OT-CFM	Yes	100.87	0.390	0.393	841.36	0.241	32

Table 1: MNIST

0\text{--}4\to 5\text{--}9

transport. We report transport cost, target-domain rate (TDR), target probability score (TPS), MNIST-feature FID, and total variation distance between predicted and real target-class distributions over digits

5,\ldots,9

. Sinkhorn hard matching is a finite-sample transductive baseline and does not define a learned global map. Among learned global maps, CyclOT achieves the best MNIST-feature FID and target class-distribution matching, while maintaining high target-domain validity.

Figure 3: MNIST $0\text{--}4\to 5\text{--}9$ transport.

Panel of Figure 4 — Figure 4: Training dynamics on the CelebA female-to-male translation task for Neural OT and CyclOT.

Method	Cost $\downarrow$	FID $\downarrow$	TDR $\uparrow$	TPS $\uparrow$	$\|\mathrm{Enrich\mbox{-}k100}-0.5\|\downarrow$	FIE $\downarrow$	MIE $\downarrow$
ICNN OT	2266.7034	305.2227	46.0%	$0.4851\pm 0.25$	0.3759	3.2M	35.2M
Neural OT	687.9927	17.8291	79.9%	$0.7447\pm 0.27$	0.0228	17.6M	1.6M
CyclOT	1069.7815	25.4478	92.0%	0.8602 $\pm$ 0.20	0.0357	2.73M	2.73M

Table 2: Quantitative comparison between Neural OT and CyclOT on the CelebA female-to-male translation task.

Input set	TPS	Predicted male	Predicted female
Real source female	0.1278 $\pm$ 0.23	479 (9.6%)	4521 (90.4%)
Real target male	0.8893 $\pm$ 0.21	4629 (92.6%)	371 (7.4%)

Table 3: FairFace classifier baseline results on real CelebA source-female and target-male images. Each evaluation set contains 5,000 images. The classifier correctly predicts 90.4% of the source-female images as female and 92.6% of the real target-male images as male.

Panel of Figure 5 — Figure 5: Cycle visualizations on CelebA. The first block shows source female samples $x$ , forward mapped samples $f_{\theta}(x)$ , and cycle reconstructions $g_{\phi}(f_{\theta}(x))$ . The second block shows source male samples $y$ , backward mapped samples $g_{\phi}(y)$ , and cycle reconstructions $f_{\theta}(g_{\phi}(y))$ .

Figure 6: Comparison results for female-to-male face translation on CelebA. All three methods are trained for 4 hours under the same experimental setting.

Figure 7: Target Domain Rate (TDR) versus training time for Neural OT and CyclOT.

Method	Transport Cost $\downarrow$	MMD $\downarrow$	$\|\mathrm{Enrich\mbox{-}k100}-0.5\|\downarrow$	Signature Corr $\uparrow$
Sinkhorn	5.2320	–	–	–
CyclOT	2.8352	0.002472	0.00729	0.99093
Neural OT	3.1925	0.002879	0.01835	0.98936
ICNN OT	3.2515	0.002623	0.00456	0.98698
Rectified flow	9.4423	0.004876	0.02044	0.94781
OT-CFM	7.0150	0.012698	0.11412	0.86522

Table 4: 4i trametinib single-cell perturbation transport.

Method	Cost $\downarrow$	$\|\mathrm{Enrich\mbox{-}k100}-0.5\|\downarrow$	TPS $\uparrow$	Training Time (hours:minutes) $\downarrow$
Sinkhorn	5643.2637	–	–	$<$ 0:01
ICNN OT	65095.5671	0.4984	0.1513 $\pm$ 0.0409	1:01
Neural OT	1984.1612	0.3246	0.4762 $\pm$ 0.1860	19:15
CyclOT	1717.2522	0.3169	0.4098 $\pm$ 0.1873	8:41

Table 5: Quantitative comparison between ICCN OT, Neural OT, and CyclOT on the chest x-ray healthy-to-pneumonia transition task.

Input set	TPS
Real source healthy	0.1876 $\pm$ 0.1382
Real target pneumonia	0.6254 $\pm$ 0.1902

Table 6: Pneumonia classifier baseline results on real source (healthy) and target (pneumonia) images.

Figure 8: Chest x-ray transport results for CyclOT. The first row shows source (healthy) radiographs $x$ , the second row shows the transported samples $f_{\theta}(x)$ , and the third row shows the cycle reconstructions $g_{\phi}(f_{\theta}(x))$ . The learned transport changes the target-domain radiographic appearance while preserving the coarse anatomical layout.

Panel of Figure 9 — Figure 9: Chest x-ray transport results for other tested methods. The first row shows source (healthy) radiographs $x$ and the second row shows the transported samples $f_{\theta}(x)$ .

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 10 — Figure 10: Additional qualitative results on the MNIST Digit Transfer (from digit 0-4 to digit 5-9) translation task.

Figure 11: Qualitative results using the first 50 female images from the CelebA dataset as fixed source samples. Odd-numbered rows show the source female images, and the corresponding even-numbered rows show the mapped male images generated by CyclOT. The full image grid is displayed without post-hoc cropping.

Figure 12: Qualitative results using the first 50 non-smiling images from the CelebA dataset as fixed source samples. Odd-numbered rows show the source non-smiling images, and the corresponding even-numbered rows show the mapped smiling images generated by CyclOT. The full image grid is displayed without post-hoc cropping.

Method	Direction	Cost $\downarrow$	$\|\mathrm{Enrich\mbox{-}k100}-0.5\|\downarrow$	TPS $\uparrow$
ICNN OT	Pos $\to$ Neg	73142.1836	0.4986	0.1517 $\pm$ 0.0260
ICNN OT	Neg $\to$ Pos	65095.5671	0.4984	0.1513 $\pm$ 0.0409
Neural OT	Pos $\to$ Neg	1623.5243	0.0507	0.4180 $\pm$ 0.1811
Neural OT	Neg $\to$ Pos	1984.1612	0.3246	0.4762 $\pm$ 0.1860
Ours	Pos $\to$ Neg	1472.9777	0.0133	0.4484 $\pm$ 0.1856
Ours	Neg $\to$ Pos	1717.2522	0.3169	0.4098 $\pm$ 0.1873

Appendix

Table 7: Quantitative comparison between ICNN OT, Neural OT, and our method on the chest x-ray healthy to pneumonia transition task, broken down by transport direction.

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Abstract

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