# Conditional Predictive Sufficient Statistics for Visual Representation Learning

arXiv: https://arxiv.org/abs/2609.30647 · HTML: https://mlsanity.com/paper/2609.30647/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 62 of its 81 references that are in the map.
- First posted on arXiv 25 Sep 2026.

## Builds on (most cited first, 30 of 62)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [DINOv2: Learning Robust Visual Features without Supervision](https://mlsanity.com/paper/2304.07193/) (2023)
- [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)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Taming Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2012.09841/) (2020)
- [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)
- [BEiT: BERT Pre-Training of Image Transformers](https://mlsanity.com/paper/2106.08254/) (2021)
- [Unsupervised Feature Learning via Non-Parametric Instance-level Discrimination](https://mlsanity.com/paper/1805.01978/) (2018)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [An Empirical Study of Training Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.02057/) (2021)
- [Unsupervised Representation Learning by Predicting Image Rotations](https://mlsanity.com/paper/1803.07728/) (2018)
- [Context Encoders: Feature Learning by Inpainting](https://mlsanity.com/paper/1604.07379/) (2016)

## Abstract (arXiv)

A useful visual representation is a statistic of the observed past that retains the latent factors shared with the future and discards patch-private noise. We formalize this requirement as a conditional predictive sufficient statistic (CPSS). Under a shared-factor model of image patches, the mutual information between the past and the next patch equals the information the past carries about the shared factor, up to a remainder that the next patch itself fails to reveal. Predicting the next patch embedding with a cosine loss is maximum likelihood for a von Mises-Fisher model of that embedding's direction, and is therefore a tractable surrogate for the predictive information. The same population loss is also minimized by a constant embedding, so stop-gradient does not by itself select the sufficient statistic; it only blocks the symmetric gradient that implements the constant solution in one step. The regression target is a shallow embedding, which forces the network output back into that shallow range and leaves the sufficient statistic in intermediate blocks. Small causal Transformers on MNIST and CIFAR-10 are used as diagnostics, not as a leaderboard. On MNIST the future shift and the stop-gradient move probe accuracy by tens of points, and the CPSS readout peaks before the output. On CIFAR-10, with the same short budget and no augmentation, every objective lands near a linear classifier on pixels. What still matches the derivation is the geometry: the CPSS output is a worse readout than its best intermediate block, next-pixel regression does not pay that penalty, and removing the stop-gradient collapses the effective rank of the embedding even when the pretext loss looks perfect.