# Hard Negative Sampling via Regularized Optimal Transport for Contrastive Representation Learning

arXiv: https://arxiv.org/abs/2111.03169 · HTML: https://mlsanity.com/paper/2111.03169/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 39 references that are in the map.
- First posted on arXiv 4 Nov 2021.

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

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [FaceNet: A Unified Embedding for Face Recognition and Clustering](https://mlsanity.com/paper/1503.03832/) (2015)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [DeepWalk: Online Learning of Social Representations](https://mlsanity.com/paper/1403.6652/) (2014)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [Approximate Nearest Neighbor Negative Contrastive Learning for Dense Text Retrieval](https://mlsanity.com/paper/2007.00808/) (2020)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Contrastive Multi-View Representation Learning on Graphs](https://mlsanity.com/paper/2006.05582/) (2020)
- [Contrastive Learning with Hard Negative Samples](https://mlsanity.com/paper/2010.04592/) (2020)
- [Deep Graph Contrastive Representation Learning](https://mlsanity.com/paper/2006.04131/) (2020)
- [A Theoretical Analysis of Contrastive Unsupervised Representation Learning](https://mlsanity.com/paper/1902.09229/) (2019)
- [InfoGraph: Unsupervised and Semi-supervised Graph-Level Representation Learning via Mutual Information Maximization](https://mlsanity.com/paper/1908.01000/) (2019)
- [Hard Negative Mixing for Contrastive Learning](https://mlsanity.com/paper/2010.01028/) (2020)
- [Debiased Contrastive Learning](https://mlsanity.com/paper/2007.00224/) (2020)
- [On Mutual Information Maximization for Representation Learning](https://mlsanity.com/paper/1907.13625/) (2019)
- [An efficient framework for learning sentence representations](https://mlsanity.com/paper/1803.02893/) (2018)
- [word2vec Explained: deriving Mikolov et al.'s negative-sampling word-embedding method](https://mlsanity.com/paper/1402.3722/) (2014)
- [Contrastive Learning with Adversarial Examples](https://mlsanity.com/paper/2010.12050/) (2020)
- [Conditional Negative Sampling for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2010.02037/) (2020)
- [Are all negatives created equal in contrastive instance discrimination?](https://mlsanity.com/paper/2010.06682/) (2020)
- [Unsupervised Hard Example Mining from Videos for Improved Object Detection](https://mlsanity.com/paper/1808.04285/) (2018)
- [Contrastive Attraction and Contrastive Repulsion for Representation Learning](https://mlsanity.com/paper/2105.03746/) (2021)
- [A Tale of Two Efficient and Informative Negative Sampling Distributions](https://mlsanity.com/paper/2012.15843/) (2020)

## Built on by (most cited first, 4 of 4)

- [Supervised Contrastive Learning with Hard Negative Samples](https://mlsanity.com/paper/2209.00078/) (2022)
- [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)
- [Hystar: Hypernetwork-driven Style-adaptive Retrieval via Dynamic SVD Modulation](https://mlsanity.com/paper/2605.10009/) (2026)

## Abstract (arXiv)

We study the problem of designing hard negative sampling distributions for unsupervised contrastive representation learning. We propose and analyze a novel min-max framework that seeks a representation which minimizes the maximum (worst-case) generalized contrastive learning loss over all couplings (joint distributions between positive and negative samples subject to marginal constraints) and prove that the resulting min-max optimum representation will be degenerate. This provides the first theoretical justification for incorporating additional regularization constraints on the couplings. We re-interpret the min-max problem through the lens of Optimal Transport (OT) theory and utilize regularized transport couplings to control the degree of hardness of negative examples. Through experiments we demonstrate that the negative samples generated from our designed negative distribution are more similar to the anchor than those generated from the baseline negative distribution. We also demonstrate that entropic regularization yields negative sampling distributions with parametric form similar to that in a recent state-of-the-art negative sampling design and has similar performance in multiple datasets. Utilizing the uncovered connection with OT, we propose a new ground cost for designing the negative distribution and show improved performance of the learned representation on downstream tasks compared to the representation learned when using squared Euclidean cost.