# A Mathematical Perspective On Contrastive Learning

arXiv: https://arxiv.org/abs/2505.24134 · HTML: https://mlsanity.com/paper/2505.24134/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 43 references that are in the map.
- First posted on arXiv 30 May 2025.

## Builds on (most cited first, 24 of 24)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Segment Anything](https://mlsanity.com/paper/2304.02643/) (2023)
- [Visual Instruction Tuning](https://mlsanity.com/paper/2304.08485/) (2023)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [Scaling Up Visual and Vision-Language Representation Learning With Noisy Text Supervision](https://mlsanity.com/paper/2102.05918/) (2021)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [Sigmoid Loss for Language Image Pre-Training](https://mlsanity.com/paper/2303.15343/) (2023)
- [Exploring Simple Siamese Representation Learning](https://mlsanity.com/paper/2011.10566/) (2020)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [CoCa: Contrastive Captioners are Image-Text Foundation Models](https://mlsanity.com/paper/2205.01917/) (2022)
- [Reproducible scaling laws for contrastive language-image learning](https://mlsanity.com/paper/2212.07143/) (2022)
- [CLIP4Clip: An Empirical Study of CLIP for End to End Video Clip Retrieval](https://mlsanity.com/paper/2104.08860/) (2021)
- [End-to-End Learning of Visual Representations from Uncurated Instructional Videos](https://mlsanity.com/paper/1912.06430/) (2019)
- [CLIP-NeRF: Text-and-Image Driven Manipulation of Neural Radiance Fields](https://mlsanity.com/paper/2112.05139/) (2021)
- [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)
- [Towards Open World Object Detection](https://mlsanity.com/paper/2103.02603/) (2021)
- [CLIP2Video: Mastering Video-Text Retrieval via Image CLIP](https://mlsanity.com/paper/2106.11097/) (2021)
- [On Mutual Information in Contrastive Learning for Visual Representations](https://mlsanity.com/paper/2005.13149/) (2020)
- [Optimal low-rank approximations of Bayesian linear inverse problems](https://mlsanity.com/paper/1407.3463/) (2014)
- [Hyperbolic Contrastive Learning](https://mlsanity.com/paper/2302.01409/) (2023)

## Built on by (most cited first, 1 of 1)

- [Expressivity In Multimodal Contrastive Learning](https://mlsanity.com/paper/2608.17203/) (2026)

## Abstract (arXiv)

Multimodal contrastive learning is a methodology for linking different data modalities; the canonical example is linking image and text data. The methodology is typically framed as the identification of a set of encoders, one for each modality, that align representations within a common latent space. In this work, we focus on the bimodal setting and interpret contrastive learning as the optimization of (parameterized) encoders that define conditional probability distributions, for each modality conditioned on the other, consistent with the available data. This provides a framework for multimodal algorithms such as crossmodal retrieval, which identifies the mode of one of these conditional distributions, and crossmodal classification, which is similar to retrieval but includes a fine-tuning step to make it task specific. The framework we adopt also gives rise to crossmodal generative models. This probabilistic perspective suggests two natural generalizations of contrastive learning: the introduction of novel probabilistic loss functions, and the use of alternative metrics for measuring alignment in the common latent space. We study these generalizations of the classical approach in the multivariate Gaussian setting. In this context we view the latent space identification as a low-rank matrix approximation problem. This allows us to characterize the capabilities of loss functions and alignment metrics to approximate natural statistics, such as conditional means and covariances; doing so yields novel variants on contrastive learning algorithms for specific mode-seeking and for generative tasks. The framework we introduce is also studied through numerical experiments on multivariate Gaussians, the labeled MNIST dataset, and on a data assimilation application arising in oceanography.