# A Unified Geometric Framework for Weighted Contrastive Learning

arXiv: https://arxiv.org/abs/2605.13943 · HTML: https://mlsanity.com/paper/2605.13943/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 31 references that are in the map.
- First posted on arXiv 13 May 2026.

## Builds on (most cited first, 22 of 22)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [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)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [Supervised Contrastive Learning](https://mlsanity.com/paper/2004.11362/) (2020)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [Learning deep representations by mutual information estimation and maximization](https://mlsanity.com/paper/1808.06670/) (2018)
- [Learning Representations by Maximizing Mutual Information Across Views](https://mlsanity.com/paper/1906.00910/) (2019)
- [On Variational Bounds of Mutual Information](https://mlsanity.com/paper/1905.06922/) (2019)
- [A Theoretical Analysis of Contrastive Unsupervised Representation Learning](https://mlsanity.com/paper/1902.09229/) (2019)
- [On Mutual Information Maximization for Representation Learning](https://mlsanity.com/paper/1907.13625/) (2019)
- [Provable Guarantees for Self-Supervised Deep Learning with Spectral Contrastive Loss](https://mlsanity.com/paper/2106.04156/) (2021)
- [Contrastive and Non-Contrastive Self-Supervised Learning Recover Global and Local Spectral Embedding Methods](https://mlsanity.com/paper/2205.11508/) (2022)
- [Learnable latent embeddings for joint behavioral and neural analysis](https://mlsanity.com/paper/2204.00673/) (2022)
- [Understanding Contrastive Learning Requires Incorporating Inductive Biases](https://mlsanity.com/paper/2202.14037/) (2022)
- [Rank-N-Contrast: Learning Continuous Representations for Regression](https://mlsanity.com/paper/2210.01189/) (2022)
- [Dissecting Supervised Contrastive Learning](https://mlsanity.com/paper/2102.08817/) (2021)
- [Contrastive Learning with Continuous Proxy Meta-Data for 3D MRI Classification](https://mlsanity.com/paper/2106.08808/) (2021)
- [A Theoretical Study of Inductive Biases in Contrastive Learning](https://mlsanity.com/paper/2211.14699/) (2022)
- [Contrastive learning for regression in multi-site brain age prediction](https://mlsanity.com/paper/2211.08326/) (2022)
- [mathbbX-Sample Contrastive Loss: Improving Contrastive Learning with Sample Similarity Graphs](https://mlsanity.com/paper/2407.18134/) (2024)
- [Supervised Contrastive Representation Learning: Landscape Analysis with Unconstrained Features](https://mlsanity.com/paper/2402.18884/) (2024)

## Abstract (arXiv)

Contrastive learning (CL) aims to preserve relational structure between samples by learning representations that reflect a similarity graph. Yet, the geometry of the resulting embeddings remains poorly understood. Here we show that weighted InfoNCE objectives can be interpreted as Distance Geometry Problems, where the weighting scheme specifies the target geometry to be realized by the representation. This viewpoint yields exact characterizations of the optimal embeddings for several supervised and weakly supervised objectives. In supervised classification, both SupCon and Soft SupCon (a dense relaxation of it where pairs from distinct classes have small non-zero similarity) collapse samples within each class to a single prototype. However, while balanced SupCon recovers the classical regular simplex geometry, class imbalance breaks this symmetry: SupCon induces non-uniform inter-class similarities depending on class sizes, whereas Soft SupCon preserves a regular simplex geometry regardless of class imbalance. In continuous-label settings, our framework reveals a different failure mode: y-Aware CL generally cannot attain its entropic optimum unless the labels lie on a hypersphere, exposing a mismatch between Euclidean label weights and spherical latent similarity. By contrast, geometrically consistent choices such as Euclidean-Euclidean weighting or X-CLR admit unique optimal embeddings. Our results show that the choice of weighting scheme determines whether contrastive learning is geometrically realizable, degenerate, or inconsistent, providing a principled framework for designing contrastive objectives.