# Theoretical Foundations of Representation Learning using Unlabeled Data: Statistics and Optimization

arXiv: https://arxiv.org/abs/2509.18997 · HTML: https://mlsanity.com/paper/2509.18997/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 51 of its 81 references that are in the map.
- First posted on arXiv 23 Sep 2025.

## Builds on (most cited first, 30 of 51)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [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)
- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [A Theoretical Analysis of Contrastive Unsupervised Representation Learning](https://mlsanity.com/paper/1902.09229/) (2019)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Explaining Neural Scaling Laws](https://mlsanity.com/paper/2102.06701/) (2021)
- [Provable Guarantees for Self-Supervised Deep Learning with Spectral Contrastive Loss](https://mlsanity.com/paper/2106.04156/) (2021)
- [Unsolved Problems in ML Safety](https://mlsanity.com/paper/2109.13916/) (2021)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Towards K-means-friendly Spaces: Simultaneous Deep Learning and Clustering](https://mlsanity.com/paper/1610.04794/) (2016)
- [Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks](https://mlsanity.com/paper/2002.02561/) (2020)
- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)
- [Demystifying Contrastive Self-Supervised Learning: Invariances, Augmentations and Dataset Biases](https://mlsanity.com/paper/2007.13916/) (2020)
- [On the linearity of large non-linear models: when and why the tangent kernel is constant](https://mlsanity.com/paper/2010.01092/) (2020)
- [Toward Understanding the Feature Learning Process of Self-supervised Contrastive Learning](https://mlsanity.com/paper/2105.15134/) (2021)
- [A Dynamical Model of Neural Scaling Laws](https://mlsanity.com/paper/2402.01092/) (2024)
- [Learning Parities with Neural Networks](https://mlsanity.com/paper/2002.07400/) (2020)
- [Tensor Programs IIb: Architectural Universality of Neural Tangent Kernel Training Dynamics](https://mlsanity.com/paper/2105.03703/) (2021)
- [Learning by Reconstruction Produces Uninformative Features For Perception](https://mlsanity.com/paper/2402.11337/) (2024)
- [Loss Landscapes of Regularized Linear Autoencoders](https://mlsanity.com/paper/1901.08168/) (2019)
- [Early Stopping in Deep Networks: Double Descent and How to Eliminate it](https://mlsanity.com/paper/2007.10099/) (2020)
- [The SSL Interplay: Augmentations, Inductive Bias, and Generalization](https://mlsanity.com/paper/2302.02774/) (2023)

## Abstract (arXiv)

Representation learning from unlabeled data has been extensively studied in statistics, data science and signal processing with a rich literature on techniques for dimension reduction, compression, multi-dimensional scaling among others. However, current deep learning models use new principles for unsupervised representation learning that cannot be easily analyzed using classical theories. For example, visual foundation models have found tremendous success using self-supervision or denoising/masked autoencoders, which effectively learn representations from massive amounts of unlabeled data. However, it remains difficult to characterize the representations learned by these models and to explain why they perform well for diverse prediction tasks or show emergent behavior. To answer these questions, one needs to combine mathematical tools from statistics and optimization. This paper provides an overview of recent theoretical advances in representation learning from unlabeled data and mentions our contributions in this direction.