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

arXiv: https://arxiv.org/abs/2609.33741 · HTML: https://mlsanity.com/paper/2609.33741/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 38 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 26 of 26)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [BLIP: Bootstrapping Language-Image Pre-training for Unified Vision-Language Understanding and Generation](https://mlsanity.com/paper/2201.12086/) (2022)
- [Scaling Up Visual and Vision-Language Representation Learning With Noisy Text Supervision](https://mlsanity.com/paper/2102.05918/) (2021)
- [Supervised Contrastive Learning](https://mlsanity.com/paper/2004.11362/) (2020)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [VICReg: Variance-Invariance-Covariance Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2105.04906/) (2021)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [Contrastive Learning with Hard Negative Samples](https://mlsanity.com/paper/2010.04592/) (2020)
- [On Mutual Information Maximization for Representation Learning](https://mlsanity.com/paper/1907.13625/) (2019)
- [Dissecting Supervised Contrastive Learning](https://mlsanity.com/paper/2102.08817/) (2021)
- [Sinkhorn Label Allocation: Semi-Supervised Classification via Annealed Self-Training](https://mlsanity.com/paper/2102.08622/) (2021)
- [A Statistical Theory of Contrastive Pre-training and Multimodal Generative AI](https://mlsanity.com/paper/2501.04641/) (2025)
- [Engineering the Neural Collapse Geometry of Supervised-Contrastive Loss](https://mlsanity.com/paper/2310.00893/) (2023)
- [Contrasting Multiple Representations with the Multi-Marginal Matching Gap](https://mlsanity.com/paper/2405.19532/) (2024)
- [Hard-Negative Sampling for Contrastive Learning: Optimal Representation Geometry and Neural- vs Dimensional-Collapse](https://mlsanity.com/paper/2311.05139/) (2023)
- [Analysis of Using Sigmoid Loss for Contrastive Learning](https://mlsanity.com/paper/2402.12613/) (2024)
- [Your contrastive learning problem is secretly a distribution alignment problem](https://mlsanity.com/paper/2502.20141/) (2025)
- [Towards understanding neural collapse in supervised contrastive learning with the information bottleneck method](https://mlsanity.com/paper/2305.11957/) (2023)
- [Symmetric Neural-Collapse Representations with Supervised Contrastive Loss: The Impact of ReLU and Batching](https://mlsanity.com/paper/2306.07960/) (2023)
- [Supervised Contrastive Representation Learning: Landscape Analysis with Unconstrained Features](https://mlsanity.com/paper/2402.18884/) (2024)
- [Global Minimizers of Sigmoid Contrastive Loss](https://mlsanity.com/paper/2509.18552/) (2025)
- [A Generalization Theory for Zero-Shot Prediction](https://mlsanity.com/paper/2507.09128/) (2025)
- [PROTOCOL: Partial Optimal Transport-enhanced Contrastive Learning for Imbalanced Multi-view Clustering](https://mlsanity.com/paper/2506.12408/) (2025)

## Abstract (arXiv)

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 textitDO 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 textittriplets. 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 textitpreviously 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.