# Understanding In-Context Learning on Structured Manifolds: Bridging Attention to Kernel Methods

arXiv: https://arxiv.org/abs/2506.10959 · HTML: https://mlsanity.com/paper/2506.10959/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 60 references that are in the map.
- First posted on arXiv 12 Jun 2025.

## Builds on (most cited first, 30 of 34)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Pre-train, Prompt, and Predict: A Systematic Survey of Prompting Methods in Natural Language Processing](https://mlsanity.com/paper/2107.13586/) (2021)
- [WorldSimBench: Towards Video Generation Models as World Simulators](https://mlsanity.com/paper/2410.18072/) (2024)
- [Transformers learn in-context by gradient descent](https://mlsanity.com/paper/2212.07677/) (2022)
- [What Can Transformers Learn In-Context? A Case Study of Simple Function Classes](https://mlsanity.com/paper/2208.01066/) (2022)
- [What learning algorithm is in-context learning? Investigations with linear models](https://mlsanity.com/paper/2211.15661/) (2022)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [Trained Transformers Learn Linear Models In-Context](https://mlsanity.com/paper/2306.09927/) (2023)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Transformers as Statisticians: Provable In-Context Learning with In-Context Algorithm Selection](https://mlsanity.com/paper/2306.04637/) (2023)
- [Transformers as Algorithms: Generalization and Stability in In-context Learning](https://mlsanity.com/paper/2301.07067/) (2023)
- [Transformer Dissection: A Unified Understanding of Transformer's Attention via the Lens of Kernel](https://mlsanity.com/paper/1908.11775/) (2019)
- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Deep ReLU network approximation of functions on a manifold](https://mlsanity.com/paper/1908.00695/) (2019)
- [Nonparametric Regression on Low-Dimensional Manifolds using Deep ReLU Networks : Function Approximation and Statistical Recovery](https://mlsanity.com/paper/1908.01842/) (2019)
- [Transformers Implement Functional Gradient Descent to Learn Non-Linear Functions In Context](https://mlsanity.com/paper/2312.06528/) (2023)
- [Provable approximation properties for deep neural networks](https://mlsanity.com/paper/1509.07385/) (2015)
- [Transformers are Minimax Optimal Nonparametric In-Context Learners](https://mlsanity.com/paper/2408.12186/) (2024)
- [Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networks](https://mlsanity.com/paper/2109.02832/) (2021)
- [Approximation and Estimation Ability of Transformers for Sequence-to-Sequence Functions with Infinite Dimensional Input](https://mlsanity.com/paper/2305.18699/) (2023)
- [Understanding Scaling Laws with Statistical and Approximation Theory for Transformer Neural Networks on Intrinsically Low-dimensional Data](https://mlsanity.com/paper/2411.06646/) (2024)
- [Understanding Emergent In-Context Learning from a Kernel Regression Perspective](https://mlsanity.com/paper/2305.12766/) (2023)
- [Distribution Approximation and Statistical Estimation Guarantees of Generative Adversarial Networks](https://mlsanity.com/paper/2002.03938/) (2020)
- [Transformers are Deep Infinite-Dimensional Non-Mercer Binary Kernel Machines](https://mlsanity.com/paper/2106.01506/) (2021)
- [Nonlocal Attention Operator: Materializing Hidden Knowledge Towards Interpretable Physics Discovery](https://mlsanity.com/paper/2408.07307/) (2024)
- [On the rate of convergence of a classifier based on a Transformer encoder](https://mlsanity.com/paper/2111.14574/) (2021)

## Built on by (most cited first, 15 of 15)

- [Pretrain-Test Task Alignment Governs Generalization in In-Context Learning](https://mlsanity.com/paper/2509.26551/) (2025)
- [Efficient and Minimax Optimal In-context Nonparametric Regression with Transformers](https://mlsanity.com/paper/2601.15014/) (2026)
- [Understanding In-Context Learning for Nonlinear Regression with Transformers: Attention as Featurizer](https://mlsanity.com/paper/2605.05176/) (2026)
- [Learn to Evolve: Self-supervised Neural JKO Operator for Wasserstein Gradient Flow](https://mlsanity.com/paper/2601.05583/) (2026)
- [Learning Theory of Transformers: Local-to-Global Approximation via Softmax Partition of Unity](https://mlsanity.com/paper/2605.08811/) (2026)
- [In-Context Optimization for Retrieval-Augmented Generation: A Gradient-Descent Perspective](https://mlsanity.com/paper/2605.26356/) (2026)
- [Beyond the Prompt in Large Language Models: Comprehension, In-Context Learning, and Chain-of-Thought](https://mlsanity.com/paper/2603.10000/) (2026)
- [Transformers Can Implement Preconditioned Richardson Iteration for In-Context Gaussian Kernel Regression](https://mlsanity.com/paper/2605.08475/) (2026)
- [Ghost in the Kernel: In-Context Learning with Efficient Transformers via Domain Generalization](https://mlsanity.com/paper/2607.00479/) (2026)
- [Sequential Correlations Change In-Context Learning: Effective Context Length and Architectural Mismatch](https://mlsanity.com/paper/2607.03660/) (2026)
- [Training-Free Universal Approximation by Prompting Random Transformers](https://mlsanity.com/paper/2608.09558/) (2026)
- [Universal Approximation of Measure-to-Measure Operators by Pushforwards](https://mlsanity.com/paper/2609.35483/) (2026)
- [In-Context Multi-Operator Learning with DeepOSets](https://mlsanity.com/paper/2512.16074/) (2025)
- [Theory on Attention Dynamics for Out-of-Distribution In-Context Learning](https://mlsanity.com/paper/2609.36448/) (2026)
- [Transformer-based In-Context Score Operator Learning for Diffusion Models](https://mlsanity.com/paper/2610.13690/) (2026)

## Abstract (arXiv)

While in-context learning (ICL) has achieved remarkable success in natural language and vision domains, its theoretical understanding-particularly in the context of structured geometric data-remains unexplored. This paper initiates a theoretical study of ICL for regression of H"older functions on manifolds. We establish a novel connection between the attention mechanism and classical kernel methods, demonstrating that transformers effectively perform kernel-based prediction at a new query through its interaction with the prompt. This connection is validated by numerical experiments, revealing that the learned query-prompt scores for H"older functions are highly correlated with the Gaussian kernel. Building on this insight, we derive generalization error bounds in terms of the prompt length and the number of training tasks. When a sufficient number of training tasks are observed, transformers give rise to the minimax regression rate of H"older functions on manifolds, which scales exponentially with respect to the prompt length with the exponent depending on the intrinsic dimension of the manifold, rather than the ambient space dimension. Our result also characterizes how the generalization error scales with the number of training tasks, shedding light on the complexity of transformers as in-context kernel algorithm learners. Our findings provide foundational insights into the role of geometry in ICL and novels tools to study ICL of nonlinear models.