# Provable Low-Frequency Bias of In-Context Learning of Representations

arXiv: https://arxiv.org/abs/2507.13540 · HTML: https://mlsanity.com/paper/2507.13540/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 33 references that are in the map.
- First posted on arXiv 17 Jul 2025.

## Builds on (most cited first, 25 of 25)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Rethinking the Role of Demonstrations: What Makes In-Context Learning Work?](https://mlsanity.com/paper/2202.12837/) (2022)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [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)
- [PAL: Program-aided Language Models](https://mlsanity.com/paper/2211.10435/) (2022)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Transformers learn to implement preconditioned gradient descent for in-context learning](https://mlsanity.com/paper/2306.00297/) (2023)
- [Why Can GPT Learn In-Context? Language Models Implicitly Perform Gradient Descent as Meta-Optimizers](https://mlsanity.com/paper/2212.10559/) (2022)
- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [In-Context Convergence of Transformers](https://mlsanity.com/paper/2310.05249/) (2023)
- [White-Box Transformers via Sparse Rate Reduction](https://mlsanity.com/paper/2306.01129/) (2023)
- [Graph Neural Networks Inspired by Classical Iterative Algorithms](https://mlsanity.com/paper/2103.06064/) (2021)
- [Asymptotic theory of in-context learning by linear attention](https://mlsanity.com/paper/2405.11751/) (2024)
- [ICLR: In-Context Learning of Representations](https://mlsanity.com/paper/2501.00070/) (2024)
- [Transformers from an Optimization Perspective](https://mlsanity.com/paper/2205.13891/) (2022)
- [Exploring the Robustness of In-Context Learning with Noisy Labels](https://mlsanity.com/paper/2404.18191/) (2024)
- [No Need for Explanations: LLMs can implicitly learn from mistakes in-context](https://mlsanity.com/paper/2502.08550/) (2025)
- [Transformers Meet In-Context Learning: A Universal Approximation Theory](https://mlsanity.com/paper/2506.05200/) (2025)
- [Recurrent Self-Attention Dynamics: An Energy-Agnostic Perspective from Jacobians](https://mlsanity.com/paper/2505.19458/) (2025)

## Built on by (most cited first, 3 of 3)

- [Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior](https://mlsanity.com/paper/2605.05115/) (2026)
- [Large language models reorganize representational geometry during in-context learning](https://mlsanity.com/paper/2605.28854/) (2026)
- [Adaptivity Under Realizability Constraints: Comparing In-Context and Agentic Learning](https://mlsanity.com/paper/2605.04995/) (2026)

## Abstract (arXiv)

In-context learning (ICL) enables large language models (LLMs) to acquire new behaviors from the input sequence alone without any parameter updates. Recent studies have shown that ICL can surpass the original meaning learned in pretraining stage through internalizing the structure the data-generating process (DGP) of the prompt into the hidden representations. However, the mechanisms by which LLMs achieve this ability is left open. In this paper, we present the first rigorous explanation of such phenomena by introducing a unified framework of double convergence, where hidden representations converge both over context and across layers. This double convergence process leads to an implicit bias towards smooth (low-frequency) representations, which we prove analytically and verify empirically. Our theory explains several open empirical observations, including why learned representations exhibit globally structured but locally distorted geometry, and why their total energy decays without vanishing. Moreover, our theory predicts that ICL has an intrinsic robustness towards high-frequency noise, which we empirically confirm. These results provide new insights into the underlying mechanisms of ICL, and a theoretical foundation to study it that hopefully extends to more general data distributions and settings.