# Learning a Flow to Self-Supervised Representations

arXiv: https://arxiv.org/abs/2609.29350 · HTML: https://mlsanity.com/paper/2609.29350/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 36 of its 41 references that are in the map.
- First posted on arXiv 24 Sep 2026.

## Builds on (most cited first, 30 of 36)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Momentum Contrast for Unsupervised Visual Representation Learning](https://mlsanity.com/paper/1911.05722/) (2019)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Exploring Simple Siamese Representation Learning](https://mlsanity.com/paper/2011.10566/) (2020)
- [Unsupervised Learning of Visual Features by Contrasting Cluster Assignments](https://mlsanity.com/paper/2006.09882/) (2020)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [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)
- [Representation Alignment for Generation: Training Diffusion Transformers Is Easier Than You Think](https://mlsanity.com/paper/2410.06940/) (2024)
- [A Theoretical Analysis of Contrastive Unsupervised Representation Learning](https://mlsanity.com/paper/1902.09229/) (2019)
- [Provable Guarantees for Self-Supervised Deep Learning with Spectral Contrastive Loss](https://mlsanity.com/paper/2106.04156/) (2021)
- [Whitening for Self-Supervised Representation Learning](https://mlsanity.com/paper/2007.06346/) (2020)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [On Feature Decorrelation in Self-Supervised Learning](https://mlsanity.com/paper/2105.00470/) (2021)
- [Towards the Generalization of Contrastive Self-Supervised Learning](https://mlsanity.com/paper/2111.00743/) (2021)
- [A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models](https://mlsanity.com/paper/2408.02320/) (2024)
- [Convergence Analysis of Probability Flow ODE for Score-based Generative Models](https://mlsanity.com/paper/2404.09730/) (2024)
- [Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances](https://mlsanity.com/paper/2401.17958/) (2024)
- [Beyond Separability: Analyzing the Linear Transferability of Contrastive Representations to Related Subpopulations](https://mlsanity.com/paper/2204.02683/) (2022)
- [Self-Supervised Flow Matching for Scalable Multi-Modal Synthesis](https://mlsanity.com/paper/2603.06507/) (2026)
- [Flow matching achieves almost minimax optimal convergence](https://mlsanity.com/paper/2405.20879/) (2024)
- [Minimax Optimality of the Probability Flow ODE for Diffusion Models](https://mlsanity.com/paper/2503.09583/) (2025)
- [An Investigation into Whitening Loss for Self-supervised Learning](https://mlsanity.com/paper/2210.03586/) (2022)
- [LDReg: Local Dimensionality Regularized Self-Supervised Learning](https://mlsanity.com/paper/2401.10474/) (2024)
- [Preventing Dimensional Collapse in Self-Supervised Learning via Orthogonality Regularization](https://mlsanity.com/paper/2411.00392/) (2024)

## Abstract (arXiv)

Explicit geometric references offer a direct way to structure self-supervised representations. Existing adversarial distribution-matching formulations, however, require costly encoder-critic optimization. We introduce Flow-Based Distribution Matching (FBDM), a non-adversarial framework that learns this reference-directed geometry through spherical conditional velocity regression. An ETF-inspired reference allows its number of components K' to exceed the auxiliary flow dimension d* while retaining structured geometric separation. We assign both augmented views of each image to the same target, while limiting how many images each reference center can receive. An explicit alignment loss further pulls the two views' representations closer together. Experiments across benchmarks ranging from CIFAR to ImageNet show that FBDM achieves performance nearly on par with DM and remains competitive with existing SSL methods. Matched training-cost comparisons show a 1.48- to 1.83-fold speedup over DM with a negligible increase in GPU memory usage. We also provide a theoretical explanation for the usefulness of the learned representations: under stated conditions, we bound the downstream misclassification rate in terms of the FBDM pretraining loss.