# Mechanism of feature learning in deep fully connected networks and kernel machines that recursively learn features

arXiv: https://arxiv.org/abs/2212.13881 · HTML: https://mlsanity.com/paper/2212.13881/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 53 of its 98 references that are in the map.
- First posted on arXiv 28 Dec 2022.

## Builds on (most cited first, 30 of 53)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [Adversarial Examples Are Not Bugs, They Are Features](https://mlsanity.com/paper/1905.02175/) (2019)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Empirical Evaluation of Rectified Activations in Convolutional Network](https://mlsanity.com/paper/1505.00853/) (2015)
- [The Pitfalls of Simplicity Bias in Neural Networks](https://mlsanity.com/paper/2006.07710/) (2020)
- [SAINT: Improved Neural Networks for Tabular Data via Row Attention and Contrastive Pre-Training](https://mlsanity.com/paper/2106.01342/) (2021)
- [Gradient Starvation: A Learning Proclivity in Neural Networks](https://mlsanity.com/paper/2011.09468/) (2020)
- [The large learning rate phase of deep learning: the catapult mechanism](https://mlsanity.com/paper/2003.02218/) (2020)
- [SGD on Neural Networks Learns Functions of Increasing Complexity](https://mlsanity.com/paper/1905.11604/) (2019)
- [Deep Information Propagation](https://mlsanity.com/paper/1611.01232/) (2016)
- [Deep learning versus kernel learning: an empirical study of loss landscape geometry and the time evolution of the Neural Tangent Kernel](https://mlsanity.com/paper/2010.15110/) (2020)
- [Dynamical Isometry and a Mean Field Theory of CNNs: How to Train 10,000-Layer Vanilla Convolutional Neural Networks](https://mlsanity.com/paper/1806.05393/) (2018)
- [The Principles of Deep Learning Theory](https://mlsanity.com/paper/2106.10165/) (2021)
- [Towards Explaining the Regularization Effect of Initial Large Learning Rate in Training Neural Networks](https://mlsanity.com/paper/1907.04595/) (2019)
- [Neural Networks can Learn Representations with Gradient Descent](https://mlsanity.com/paper/2206.15144/) (2022)
- [Linearized two-layers neural networks in high dimension](https://mlsanity.com/paper/1904.12191/) (2019)
- [The Low-Rank Simplicity Bias in Deep Networks](https://mlsanity.com/paper/2103.10427/) (2021)
- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)

## Built on by (most cited first, 25 of 25)

- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Emergence in non-neural models: grokking modular arithmetic via average gradient outer product](https://mlsanity.com/paper/2407.20199/) (2024)
- [Provable Guarantees for Neural Networks via Gradient Feature Learning](https://mlsanity.com/paper/2310.12408/) (2023)
- [Adaptive kernel predictors from feature-learning infinite limits of neural networks](https://mlsanity.com/paper/2502.07998/) (2025)
- [Residual Connections Harm Generative Representation Learning](https://mlsanity.com/paper/2404.10947/) (2024)
- [Formation of Representations in Neural Networks](https://mlsanity.com/paper/2410.03006/) (2024)
- [Faithful and Efficient Explanations for Neural Networks via Neural Tangent Kernel Surrogate Models](https://mlsanity.com/paper/2305.14585/) (2023)
- [To Grok Grokking: Provable Grokking in Ridge Regression](https://mlsanity.com/paper/2601.19791/) (2026)
- [Bridging Lottery Ticket and Grokking: Understanding Grokking from Inner Structure of Networks](https://mlsanity.com/paper/2310.19470/) (2023)
- [Understanding the Evolution of the Neural Tangent Kernel at the Edge of Stability](https://mlsanity.com/paper/2507.12837/) (2025)
- [Steering Autoregressive Music Generation with Recursive Feature Machines](https://mlsanity.com/paper/2510.19127/) (2025)
- [Mechanist: AI as a Scientific Instrument for Discovering the Mechanisms of Intelligence](https://mlsanity.com/paper/2608.12036/) (2026)
- [TrIM: Transformed Iterative Mondrian Forests for Gradient-based Dimension Reduction and High-Dimensional Regression](https://mlsanity.com/paper/2407.09964/) (2024)
- [Learning Hierarchical Polynomials of Multiple Nonlinear Features with Three-Layer Networks](https://mlsanity.com/paper/2411.17201/) (2024)
- [Local EGOP for Continuous Index Learning](https://mlsanity.com/paper/2601.07061/) (2026)
- [Gradient Flow Through Diagram Expansions: Learning Regimes and Explicit Solutions](https://mlsanity.com/paper/2602.04548/) (2026)
- [HarmThoughts: A Benchmark for Fine-Grained Harmful Behavior Detection in Reasoning Traces](https://mlsanity.com/paper/2604.19001/) (2026)
- [Impact of Label Noise on Learning Complex Features](https://mlsanity.com/paper/2411.04569/) (2024)
- [Exploring Grokking: Experimental and Mechanistic Investigations](https://mlsanity.com/paper/2412.10898/) (2024)
- [Towards a Statistical Understanding of Neural Networks: Beyond the Neural Tangent Kernel Theories](https://mlsanity.com/paper/2412.18756/) (2024)
- [Explainable Machine Learning for Oxygen Diffusion in Perovskites and Pyrochlores](https://mlsanity.com/paper/2505.11722/) (2025)
- [AGOP-IxG: A Gradient Covariance Filter for Local Feature Attribution on Tabular Data, with a Controlled Benchmark](https://mlsanity.com/paper/2605.15700/) (2026)
- [The Model Shape Behind Scaling Laws: A Gradient Superposition Perspective](https://mlsanity.com/paper/2605.27989/) (2026)
- [Fast Multi-dimensional Refusal Subspaces via RFM-AGOP](https://mlsanity.com/paper/2607.02396/) (2026)
- [Can Representation Learning Decouple from Loss Minimization? Polar Updates Have an Answer](https://mlsanity.com/paper/2609.36240/) (2026)

## Abstract (arXiv)

In recent years neural networks have achieved impressive results on many technological and scientific tasks. Yet, the mechanism through which these models automatically select features, or patterns in data, for prediction remains unclear. Identifying such a mechanism is key to advancing performance and interpretability of neural networks and promoting reliable adoption of these models in scientific applications. In this paper, we identify and characterize the mechanism through which deep fully connected neural networks learn features. We posit the Deep Neural Feature Ansatz, which states that neural feature learning occurs by implementing the average gradient outer product to up-weight features strongly related to model output. Our ansatz sheds light on various deep learning phenomena including emergence of spurious features and simplicity biases and how pruning networks can increase performance, the "lottery ticket hypothesis." Moreover, the mechanism identified in our work leads to a backpropagation-free method for feature learning with any machine learning model. To demonstrate the effectiveness of this feature learning mechanism, we use it to enable feature learning in classical, non-feature learning models known as kernel machines and show that the resulting models, which we refer to as Recursive Feature Machines, achieve state-of-the-art performance on tabular data.