Multi-Marginal Inverse Optimal Transport for Contrastive Learning Via Explicit Anchor-Positive-Negative Coupling

Ngoc-Hai Nguyen; Thuan Nguyen; Prakash Ishwar; Shuchin Aeron

stat.MLSep 27, 2026

Multi-Marginal Inverse Optimal Transport for Contrastive Learning Via Explicit Anchor-Positive-Negative Coupling

Authors: Ngoc-Hai Nguyen, Thuan Nguyen, Prakash Ishwar, Shuchin Aeron

Organizations: Department of Electrical and Computer Engineering Tufts University · Department of Engineering, Engineering Technology East Tennessee State University · Department of Electrical and Computer Engineering Boston University

Abstract

Inverse Optimal Transport (OT) based methods for representation learning learn representations such that the global OT coupling between a pair of data marginals in the representation space, concentrates on the positive pairs. This is in contrast to previous methods that primarily focused on pairwise matching. However, these methods $\textit{DO NOT}$ utilize negative pairs and hence are not truly contrastive in their approach. We show that this leads to issues of dimensional collapse and hence degraded downstream performance. To alleviate this, we develop a novel multi-marginal (MM) inverse OT (IOT) contrastive learning (CL) approach called Neg-MMIOT-CL, which learns representations such that the global multi-marginal OT (MMOT) coupling between a triple of data marginals, with respect to a carefully designed ground-cost between triplets of data points in the representation space, concentrates on the anchor-positive-negative $\textit{triplets}$ . For a latent class model, we empirically show that Neg-MMIOT-CL alleviates dimensional collapse. Furthermore, for a specific choice of ground cost for all triplets in representation space, we prove that the optimal representation configuration for Neg-MMIOT-CL exhibits equiangular property for within-class and across-class representations, which translates to Neural-Collapse when the representation dimension is larger than the number of classes minus one -- a result that is $\textit{previously established only}$ for pairwise contrastive learning methods. Finally, we propose Neg-IOT-CL-PushPull, that is a computationally efficient alternative to Neg-MMIOT-CL, alleviating the high cost of computing MMOT plans needed during implementation. We apply these methods on both synthetic and real-world datasets and show significant improvements over existing OT-based contrastive learning methods.

Figures & tables

\tilde{P}_{ijk}=\frac{\mathbf{1}\!\left((i,j,k)\in\mathcal{A}\right)}{|\mathcal{A}|}.

Algorithm 1 Neg-MMIOT-CL and Neg-IOT-CL-PushPull batch training

Figure 1: Testing Neural Collapse: NC1 measures within-class variability with smaller values indicating stronger class collapse. NC2 measures how close the class means are to the simplex-ETF geometry. Both NC2 quantities should approach zero when the class means form a simplex ETF. The numbers in the legends are the NC metrics in the last iteration.

Panel of Figure 2 — Figure 2: Testing Dimensional Collapse. We plot the sorted (max)-normalized eigenvalues of the centered class-mean covariance matrix at the final epoch. A rapidly decaying spectrum indicates that the class means occupy only a few effective directions, i.e., dimensional collapse.

Setting	Method	MNIST		SVHN		CIFAR-10		CIFAR-100		Tiny-ImageNet
		ResNet50	ViT-B/16	ResNet50	ViT-B/16	ResNet50	ViT-B/16	ResNet50	ViT-B/16	ResNet50	ViT-B/16
SCL	InfoNCE	99.36	99.49	91.96	94.07	91.39	90.40	74.01	73.59	62.60	46.32
	InvaSpread	98.98	99.61	81.79	85.35	78.46	72.62	63.26	67.08	55.06	44.19
	Standard OT	99.21	99.60	92.08	94.24	90.78	84.72	72.71	73.49	61.20	45.27
	Neg-IOT-CL-PushPull (Ours)	99.30	99.64	91.89	94.38	89.59	88.69	74.80	74.67	62.04	46.54
	Neg-MMIOT-CL (Ours)	99.41	99.66	92.74	94.87	90.64	91.97	75.17	74.62	63.54	48.02

Table 1: Average linear-probe accuracy (%) for SCL and UCL across datasets using ResNet50 and ViT-B/16 backbones. Bold values indicate the best result within each setting, dataset, and backbone.

CLIP-Loss	Image $\rightarrow$ Text		Text $\rightarrow$ Image		CIFAR-10		CIFAR-100
	Top-1	Top-5	Top-1	Top-5	Top-1	Top-5	Top-1	Top-5
infoNCE	67.22	92.51	65.91	94.61	23.37	74.82	6.36	27.59
Standard OT	43.92	81.86	30.90	70.54	26.62	78.56	4.77	22.95
Standard OT - Uniform	64.25	92.40	63.22	92.06	25.97	78.67	6.51	27.68
DBOT	15.89	46.68	15.52	45.92	24.98	70.36	5.72	27.19
Fused-Gromov	14.32	43.92	13.94	42.87	24.99	76.06	3.72	19.40

Table 2: MS-COCO retrieval and zero-shot transfer results. Bold values indicate the best result

Appendix figures & tables21 assets

Supplementary material from the paper’s appendix.

Appendix

\tilde{P}_{ijk}=\begin{cases}1,&(i,j,k)\in\mathcal{A}\\ 0,&\text{otherwise}\end{cases}

Appendix

Algorithm 2 Supervised CL with Neg-MMIOT-CL

(\tilde{P}_{+})_{ij}=\begin{cases}1,&(i,j)\in\mathcal{A}^{+}\\ 0,&\text{otherwise}\end{cases}\qquad(\tilde{P}_{-})_{ik}\!=\!=\begin{cases}1,&(i,k)\in\mathcal{A}^{-}\\ 0,&\text{otherwise}\end{cases}

Appendix

Algorithm 3 Supervised CL with Neg-IOT-CL-PushPull

\mathcal{B}^{(t)}=\{(x_{i},y_{i}):i\in\bigcup_{c=1}^{K}\mathcal{S}_{\mathfrak{c}}^{(t)}\}.

Appendix

Algorithm 4 Class-Uniform Batch Sampler (only for Synthetic Experiments under the Theoretical Setting)

Figure 3: Visualization on $\mathbb{R}^{2}$ . We use tSNE method for projection.

Figure 4: Visualization on $\mathbb{R}^{2}$ . We use tSNE method for projection.

Figure 5: Neural Collapse Observation when $\psi$ is negative log sigmoid

Panel of Figure 6 — Figure 6: Dimensional Collapse in the settings of different numbers of classes when $\psi$ is negative log sigmoid

SCL	MNIST
	ResNet18		ResNet34		ResNet50		ViT-B/16
	Linear	k-NN	Linear	k-NN	Linear	k-NN	Linear	k-NN
infoNCE	$99.25\pm 0.10$	$99.24\pm 0.13$	$99.27\pm 0.04$	$99.27\pm 0.04$	$99.36\pm 0.07$	$99.30\pm 0.12$	$99.49\pm 0.13$	$99.46\pm 0.10$
InvaSpread	$99.27\pm 0.10$	$99.03\pm 0.13$	$98.91\pm 0.08$	$98.76\pm 0.13$	$98.98\pm 0.09$	$99.17\pm 0.07$	$99.61\pm 0.08$	$99.59\pm 0.03$
Standard OT	$99.22\pm 0.04$	$99.20\pm 0.05$	$99.18\pm 0.10$	$99.18\pm 0.13$	$99.21\pm 0.04$	$99.17\pm 0.04$	$99.60\pm 0.05$	$99.58\pm 0.09$
Neg-IOT-CL-PushPull (Ours)	$\mathbf{99.33\pm 0.06}$	$\mathbf{99.35\pm 0.10}$	$\mathbf{99.33\pm 0.11}$	$\mathbf{99.31\pm 0.10}$	$99.30\pm 0.13$	$99.29\pm 0.07$	$99.64\pm 0.10$	$99.64\pm 0.05$

Appendix

Table 3: SCL results on MNIST with different backbones.

SCL	SVHN
	ResNet18		ResNet34		ResNet50		ViT-B/16
	Linear	k-NN	Linear	k-NN	Linear	k-NN	Linear	k-NN
infoNCE	$91.76\pm 0.33$	$91.71\pm 0.21$	$\mathbf{91.73\pm 0.27}$	$\mathbf{91.77\pm 0.43}$	$91.96\pm 0.42$	$91.94\pm 0.29$	$94.07\pm 0.38$	$94.04\pm 0.37$
InvaSpread	$86.55\pm 0.28$	$87.21\pm 0.36$	$82.65\pm 0.34$	$85.94\pm 0.23$	$81.79\pm 0.30$	$80.80\pm 0.30$	$85.35\pm 0.45$	$84.78\pm 0.48$
Standard OT	$91.24\pm 0.32$	$91.16\pm 0.18$	$91.58\pm 0.13$	$91.61\pm 0.15$	$92.08\pm 0.25$	$91.99\pm 0.31$	$94.24\pm 0.47$	$94.12\pm 0.36$
Neg-IOT-CL-PushPull (Ours)	$91.03\pm 0.14$	$90.99\pm 0.27$	$91.55\pm 0.27$	$91.56\pm 0.13$	$91.89\pm 0.14$	$91.83\pm 0.41$	$94.38\pm 0.33$	$94.38\pm 0.35$

Appendix

Table 4: SCL results on SVHN with different backbones.

SCL	CIFAR-10
	ResNet18		ResNet34		ResNet50		ViT-B/16
	Linear	k-NN	Linear	k-NN	Linear	k-NN	Linear	k-NN
infoNCE	$85.01\pm 0.25$	$84.82\pm 0.47$	$86.21\pm 0.27$	$86.06\pm 0.29$	$\mathbf{91.39\pm 0.34}$	$\mathbf{90.98\pm 0.43}$	$90.40\pm 0.38$	$90.32\pm 0.46$
InvaSpread	$73.01\pm 0.42$	$76.01\pm 0.49$	$75.24\pm 0.33$	$75.59\pm 0.31$	$78.46\pm 0.40$	$77.53\pm 0.47$	$72.62\pm 0.24$	$72.41\pm 0.56$
Standard OT	$83.76\pm 0.30$	$83.47\pm 0.34$	$83.29\pm 0.24$	$84.40\pm 0.41$	$90.78\pm 0.55$	$90.80\pm 0.48$	$84.72\pm 0.29$	$84.55\pm 0.57$
Neg-IOT-CL-PushPull (Ours)	$82.15\pm 0.20$	$82.66\pm 0.54$	$83.23\pm 0.20$	$82.21\pm 0.36$	$89.59\pm 0.55$	$89.62\pm 0.55$	$88.69\pm 0.53$	$88.54\pm 0.16$

Appendix

Table 5: SCL results on CIFAR-10 with different backbones.

SCL	CIFAR-100
	ResNet18		ResNet34		ResNet50		ViT-B/16
	Linear	k-NN	Linear	k-NN	Linear	k-NN	Linear	k-NN
infoNCE	$69.72\pm 0.58$	$69.46\pm 0.39$	$70.61\pm 0.33$	$70.33\pm 0.50$	$74.01\pm 0.35$	$73.56\pm 0.40$	$73.59\pm 0.33$	$73.58\pm 0.34$
InvaSpread	$55.46\pm 0.53$	$54.32\pm 0.81$	$64.84\pm 0.62$	$63.10\pm 0.46$	$63.26\pm 0.81$	$62.24\pm 0.34$	$67.08\pm 0.49$	$66.68\pm 0.30$
Standard OT	$69.24\pm 0.56$	$69.20\pm 0.58$	$70.53\pm 0.82$	$70.41\pm 0.72$	$72.71\pm 0.30$	$72.88\pm 0.34$	$73.49\pm 0.56$	$73.59\pm 0.70$
Neg-IOT-CL-PushPull (Ours)	$69.26\pm 0.29$	$69.46\pm 0.52$	$70.76\pm 0.76$	$70.56\pm 0.74$	$74.80\pm 0.75$	$74.19\pm 0.30$	$\mathbf{74.67\pm 0.69}$	$\mathbf{74.83\pm 0.67}$

Appendix

Table 6: SCL results on CIFAR-100 with different backbones.

SCL	Tiny-ImageNet
	ResNet18		ResNet34		ResNet50		ViT-B/16
	Linear	k-NN	Linear	k-NN	Linear	k-NN	Linear	k-NN
infoNCE	$54.56\pm 0.65$	$54.34\pm 0.55$	$55.02\pm 0.60$	$54.52\pm 0.54$	$62.60\pm 0.62$	$\mathbf{63.30\pm 1.00}$	$46.32\pm 0.86$	$46.37\pm 0.68$
InvaSpread	$51.84\pm 0.84$	$51.26\pm 0.42$	$51.20\pm 0.44$	$49.98\pm 0.38$	$55.06\pm 0.67$	$53.72\pm 0.56$	$44.19\pm 0.91$	$43.33\pm 0.82$
Standard OT	$54.64\pm 0.86$	$53.92\pm 0.61$	$55.44\pm 0.36$	$55.08\pm 0.92$	$61.20\pm 0.83$	$60.84\pm 0.36$	$45.27\pm 0.80$	$45.19\pm 0.41$
Neg-IOT-CL-PushPull (Ours)	$\mathbf{57.28\pm 0.74}$	$\mathbf{56.88\pm 0.47}$	$\mathbf{56.74\pm 0.80}$	$\mathbf{56.38\pm 0.79}$	$62.04\pm 0.62$	$61.42\pm 1.02$	$46.54\pm 0.54$	$46.58\pm 0.80$

Appendix

Table 7: SCL results on Tiny-ImageNet with different backbones.

UCL	MNIST
	ResNet18		ResNet34		ResNet50		ViT-B/16
	Linear	k-NN	Linear	k-NN	Linear	k-NN	Linear	k-NN
infoNCE	$97.63\pm 0.08$	$98.56\pm 0.05$	$97.43\pm 0.09$	$98.22\pm 0.05$	$97.73\pm 0.08$	$98.20\pm 0.05$	$98.40\pm 0.16$	$98.82\pm 0.12$
InvaSpread	$98.00\pm 0.18$	$97.37\pm 0.18$	$97.42\pm 0.05$	$96.93\pm 0.10$	$97.65\pm 0.15$	$96.89\pm 0.06$	$97.95\pm 0.14$	$97.70\pm 0.13$
Standard OT	$97.70\pm 0.09$	$98.07\pm 0.06$	$\mathbf{97.60\pm 0.13}$	$\mathbf{98.39\pm 0.18}$	$97.20\pm 0.06$	$98.09\pm 0.11$	$98.35\pm 0.05$	$98.74\pm 0.13$
Neg-IOT-CL-PushPull (Ours)	$\mathbf{97.94\pm 0.11}$	$\mathbf{98.62\pm 0.08}$	$97.49\pm 0.13$	$98.31\pm 0.05$	$96.56\pm 0.14$	$98.07\pm 0.13$	$98.61\pm 0.14$	$98.91\pm 0.12$

Appendix

Table 8: UCL results on MNIST with different backbones.

UCL	SVHN
	ResNet18		ResNet34		ResNet50		ViT-B/16
	Linear	k-NN	Linear	k-NN	Linear	k-NN	Linear	k-NN
infoNCE	$77.44\pm 0.60$	$81.21\pm 0.39$	$77.70\pm 0.46$	$81.86\pm 0.63$	$78.13\pm 0.36$	$83.00\pm 0.43$	$84.67\pm 0.57$	$86.24\pm 0.21$
InvaSpread	$74.36\pm 0.53$	$71.49\pm 0.37$	$72.78\pm 0.46$	$69.47\pm 0.47$	$73.48\pm 0.63$	$70.53\pm 0.38$	$75.13\pm 0.38$	$73.88\pm 0.33$
Standard OT	$74.32\pm 0.29$	$79.42\pm 0.59$	$75.65\pm 0.52$	$80.12\pm 0.61$	$75.96\pm 0.37$	$79.69\pm 0.38$	$83.94\pm 0.33$	$85.76\pm 0.23$
Neg-IOT-CL-PushPull (Ours)	$79.28\pm 0.25$	$81.79\pm 0.44$	$79.42\pm 0.43$	$81.38\pm 0.34$	$78.63\pm 0.55$	$80.85\pm 0.38$	$87.45\pm 0.30$	$87.02\pm 0.26$

Appendix

Table 9: UCL results on SVHN with different backbones.

UCL	CIFAR-10
	ResNet18		ResNet34		ResNet50		ViT-B/16
	Linear	k-NN	Linear	k-NN	Linear	k-NN	Linear	k-NN
infoNCE	$75.30\pm 0.65$	$76.00\pm 0.57$	$75.45\pm 0.30$	$76.47\pm 0.51$	$76.81\pm 0.52$	$76.69\pm 0.62$	$84.57\pm 0.53$	$85.84\pm 0.61$
InvaSpread	$67.27\pm 0.45$	$65.83\pm 0.74$	$65.92\pm 0.38$	$64.54\pm 0.65$	$64.88\pm 0.52$	$63.69\pm 0.30$	$64.13\pm 0.49$	$63.28\pm 0.64$
Standard OT	$75.33\pm 0.62$	$75.27\pm 0.34$	$75.94\pm 0.72$	$76.77\pm 0.62$	$78.12\pm 0.57$	$77.67\pm 0.35$	$83.22\pm 0.43$	$86.28\pm 0.35$
Neg-IOT-CL-PushPull (Ours)	$\mathbf{77.43\pm 0.25}$	$\mathbf{76.05\pm 0.52}$	$\mathbf{77.02\pm 0.65}$	$\mathbf{76.34\pm 0.77}$	$\mathbf{78.18\pm 0.63}$	$\mathbf{77.89\pm 0.29}$	$\mathbf{87.56\pm 0.23}$	$\mathbf{86.96\pm 0.36}$

Appendix

Table 10: UCL results on CIFAR-10 with different backbones.

UCL	CIFAR-100
	ResNet18		ResNet34		ResNet50		ViT-B/16
	Linear	k-NN	Linear	k-NN	Linear	k-NN	Linear	k-NN
infoNCE	$48.34\pm 0.32$	$49.16\pm 0.31$	$48.09\pm 0.44$	$48.60\pm 0.48$	$49.05\pm 0.75$	$49.51\pm 0.79$	$56.99\pm 0.89$	$55.19\pm 0.37$
InvaSpread	$40.37\pm 0.77$	$40.63\pm 0.82$	$44.23\pm 0.57$	$45.45\pm 0.55$	$44.51\pm 0.44$	$45.83\pm 0.44$	$44.17\pm 0.49$	$39.59\pm 0.36$
Standard OT	$46.15\pm 0.38$	$46.04\pm 0.31$	$46.35\pm 0.88$	$45.54\pm 0.41$	$45.54\pm 0.91$	$45.60\pm 0.84$	$55.46\pm 0.62$	$54.21\pm 0.35$
Neg-IOT-CL-PushPull (Ours)	$\mathbf{51.81\pm 0.79}$	$\mathbf{48.41\pm 0.73}$	$\mathbf{51.04\pm 0.40}$	$\mathbf{48.13\pm 0.44}$	$\mathbf{52.20\pm 0.71}$	$\mathbf{49.13\pm 0.52}$	$55.45\pm 0.88$	$54.52\pm 0.91$

Appendix

Table 11: UCL results on CIFAR-100 with different backbones.

UCL	Tiny-ImageNet
	ResNet18		ResNet34		ResNet50		ViT-B/16
	Linear	k-NN	Linear	k-NN	Linear	k-NN	Linear	k-NN
infoNCE	$53.68\pm 0.69$	$50.35\pm 0.44$	$50.90\pm 0.96$	$48.34\pm 0.77$	$55.23\pm 1.18$	$53.56\pm 0.92$	$51.45\pm 0.46$	$49.88\pm 0.48$
InvaSpread	$48.20\pm 0.92$	$46.61\pm 0.86$	$46.14\pm 0.76$	$42.51\pm 1.15$	$46.85\pm 1.08$	$43.51\pm 0.62$	$43.18\pm 0.52$	$39.66\pm 1.15$
Standard OT	$50.45\pm 0.97$	$46.68\pm 0.88$	$51.78\pm 0.47$	$48.75\pm 1.12$	$55.29\pm 0.70$	$52.92\pm 1.15$	$53.12\pm 0.43$	$51.44\pm 0.79$
Neg-IOT-CL-PushPull (Ours)	$50.88\pm 0.47$	$47.77\pm 0.66$	$51.70\pm 1.18$	$48.56\pm 0.96$	$55.37\pm 0.86$	$52.17\pm 0.65$	$54.35\pm 1.02$	$52.90\pm 1.06$

Appendix

Table 12: UCL results on Tiny-ImageNet with different backbones.

Learning Rate	Linear	k-NN
$5\times 10^{-4}$	79.52	79.40
$1\times 10^{-3}$	82.00	81.92
$5\times 10^{-3}$	82.15	82.66
0.01	80.00	79.85
0.03	77.70	77.81
0.1	75.95	75.04

Appendix

Table 13: Performance on CIFAR-10 of Neg-IOT-CL-PushPull under different hyperparameter settings using ResNet-18 as the backbone in SCL.

Panel of Figure 7 — Figure 7: Efficiency and scaling comparison of OT-based objectives. Left : linear-probe accuracy versus wall-clock seconds per training step for different batch sizes. Points closer to the upper-left indicate a better efficiency–performance tradeoff. Right : peak GPU memory as a function of batch size. Neg-IOT-CL-PushPull stays close to Standard OT in runtime while avoiding the steep runtime and memory growth of Neg-MMIOT-CL

	CIFAR-100				Tiny-ImageNet
SCL	ResNet18	ResNet34	ResNet50	ViT-B/16	ResNet18	ResNet34	ResNet50	ViT-B/16
InfoNCE	3.22	4.40	5.66	13.23	6.44	9.37	11.08	27.68
Standard OT	3.40	4.86	6.46	13.59	6.92	9.35	11.56	28.39
InvaSpread	3.16	4.48	5.85	13.02	6.87	9.49	11.81	28.20
OT-PushPull	3.69	5.62	7.04	13.87	7.69	10.19	12.73	29.64
MMIOT	6.01	7.04	9.64	16.71	10.25	13.39	16.05	35.19

Appendix

Table 14: Average training time (seconds per epoch) for SCL methods on CIFAR-100 and Tiny-ImageNet.

	CIFAR-100				Tiny-ImageNet
UCL	ResNet18	ResNet34	ResNet50	ViT-B/16	ResNet18	ResNet34	ResNet50	ViT-B/16
InfoNCE	71.76	74.05	75.55	86.44	244.79	250.85	277.16	288.26
Standard OT	72.38	76.26	76.37	86.80	231.37	282.39	351.46	286.67
InvaSpread	75.21	80.41	83.78	87.61	257.83	324.82	360.12	287.85
OT-PushPull	73.54	76.28	77.34	87.89	294.22	283.76	351.88	288.30
MMIOT	79.28	82.51	82.30	100.57	273.87	294.03	376.97	300.93

Appendix

Table 15: Average training time (seconds per epoch) for UCL methods on CIFAR-100 and Tiny-ImageNet.

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Contrastive learning is a successful paradigm for learning

d

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i

is closer to

j

than to

k

.'' Despite its success, understanding why contrastive learning leads to representations of high \textit{generalization} quality---beyond the often pessimistic predictions from PAC-learning---remains a central question. Recently, \citet*{alon2024optimal} proved that, for PAC-learning

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n

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Θ(\min(nd, n^2))

triplets are necessary and sufficient, while they posed as an open question whether their VC dimension bounds for the more realistic setting of \textit{contrastive learning with a margin} can be improved. For a margin parameter

α>0

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(i,j^{+},k^{-})_α

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φ:[n]\rightarrow \mathbb{R}^{d}

, if

\|φ(i)-φ(k)\|_2>(1+α)\cdot\|φ(i)-φ(j)\|_2

. In this work, we resolve their question by proving that the VC dimension of contrastive learning under any margin

α\in(0,1)

is in fact

O(n/α^2)

, improving on the previous bound of

O(n\log(n)/α^2)

. We also establish that the bounds are optimal up to constant factors, by providing a matching lower bound of

Ω(\frac{n}{α^2})

(the previously known lower bound was

Ω(\frac{n}α)

), for

α\geq \max(n^{-1/2},d^{-1/2})

Dionysis Arvanitakis, Vaggos Chatziafratis, Yiyuan Luo +1

Northwestern University · UC Santa Cruz

Contrastive Learning Voice Conversion

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