# How Spurious Features Are Memorized: Precise Analysis for Random and NTK Features

arXiv: https://arxiv.org/abs/2305.12100 · HTML: https://mlsanity.com/paper/2305.12100/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 64 references that are in the map.
- First posted on arXiv 20 May 2023.

## Builds on (most cited first, 30 of 50)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [ImageNet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness](https://mlsanity.com/paper/1811.12231/) (2018)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [What Neural Networks Memorize and Why: Discovering the Long Tail via Influence Estimation](https://mlsanity.com/paper/2008.03703/) (2020)
- [Does Learning Require Memorization? A Short Tale about a Long Tail](https://mlsanity.com/paper/1906.05271/) (2019)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [The Pitfalls of Simplicity Bias in Neural Networks](https://mlsanity.com/paper/2006.07710/) (2020)
- [Noise or Signal: The Role of Image Backgrounds in Object Recognition](https://mlsanity.com/paper/2006.09994/) (2020)
- [An Investigation of Why Overparameterization Exacerbates Spurious Correlations](https://mlsanity.com/paper/2005.04345/) (2020)
- [Deep learning: a statistical viewpoint](https://mlsanity.com/paper/2103.09177/) (2021)
- [SGD on Neural Networks Learns Functions of Increasing Complexity](https://mlsanity.com/paper/1905.11604/) (2019)
- [Stolen Memories: Leveraging Model Memorization for Calibrated White-Box Membership Inference](https://mlsanity.com/paper/1906.11798/) (2019)
- [Linearized two-layers neural networks in high dimension](https://mlsanity.com/paper/1904.12191/) (2019)
- [A Universal Law of Robustness via Isoperimetry](https://mlsanity.com/paper/2105.12806/) (2021)
- [Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation](https://mlsanity.com/paper/2105.14368/) (2021)
- [Theoretical insights into the optimization landscape of over-parameterized shallow neural networks](https://mlsanity.com/paper/1707.04926/) (2017)
- [What shapes feature representations? Exploring datasets, architectures, and training](https://mlsanity.com/paper/2006.12433/) (2020)
- [Learning curves of generic features maps for realistic datasets with a teacher-student model](https://mlsanity.com/paper/2102.08127/) (2021)
- [Salient ImageNet: How to discover spurious features in Deep Learning?](https://mlsanity.com/paper/2110.04301/) (2021)
- [The Neural Tangent Kernel in High Dimensions: Triple Descent and a Multi-Scale Theory of Generalization](https://mlsanity.com/paper/2008.06786/) (2020)
- [Algorithmic Stability for Adaptive Data Analysis](https://mlsanity.com/paper/1511.02513/) (2015)
- [Overparameterized Nonlinear Learning: Gradient Descent Takes the Shortest Path?](https://mlsanity.com/paper/1812.10004/) (2018)
- [Tight Bounds on the Smallest Eigenvalue of the Neural Tangent Kernel for Deep ReLU Networks](https://mlsanity.com/paper/2012.11654/) (2020)
- [The Interpolation Phase Transition in Neural Networks: Memorization and Generalization under Lazy Training](https://mlsanity.com/paper/2007.12826/) (2020)
- [A Random Matrix Approach to Neural Networks](https://mlsanity.com/paper/1702.05419/) (2017)
- [Spectra of the Conjugate Kernel and Neural Tangent Kernel for linear-width neural networks](https://mlsanity.com/paper/2005.11879/) (2020)

## Built on by (most cited first, 11 of 11)

- [A Theory of Non-Linear Feature Learning with One Gradient Step in Two-Layer Neural Networks](https://mlsanity.com/paper/2310.07891/) (2023)
- [The Clever Hans Mirage: A Comprehensive Survey on Spurious Correlations in Machine Learning](https://mlsanity.com/paper/2402.12715/) (2024)
- [The Pitfalls of Memorization: When Memorization Hurts Generalization](https://mlsanity.com/paper/2412.07684/) (2024)
- [Theoretical Analysis of Robust Overfitting for Wide DNNs: An NTK Approach](https://mlsanity.com/paper/2310.06112/) (2023)
- [NTK-Guided Few-Shot Class Incremental Learning](https://mlsanity.com/paper/2403.12486/) (2024)
- [Towards Understanding the Word Sensitivity of Attention Layers: A Study via Random Features](https://mlsanity.com/paper/2402.02969/) (2024)
- [ShortcutProbe: Probing Prediction Shortcuts for Learning Robust Models](https://mlsanity.com/paper/2505.13910/) (2025)
- [A Law of Data Reconstruction for Random Features (and Beyond)](https://mlsanity.com/paper/2509.22214/) (2025)
- [Review of Hallucination Understanding in Large Language and Vision Models](https://mlsanity.com/paper/2510.00034/) (2025)
- [Feature Learning in Linear-Width Two-Layer Networks: Two vs. One Step of Gradient Descent](https://mlsanity.com/paper/2605.17767/) (2026)
- [Mitigating Spurious Correlations with Memorization-Guided Dataset De-Biasing](https://mlsanity.com/paper/2606.02830/) (2026)

## Abstract (arXiv)

Deep learning models are known to overfit and memorize spurious features in the training dataset. While numerous empirical studies have aimed at understanding this phenomenon, a rigorous theoretical framework to quantify it is still missing. In this paper, we consider spurious features that are uncorrelated with the learning task, and we provide a precise characterization of how they are memorized via two separate terms: (i) the stability of the model with respect to individual training samples, and (ii) the feature alignment between the spurious feature and the full sample. While the first term is well established in learning theory and it is connected to the generalization error in classical work, the second one is, to the best of our knowledge, novel. Our key technical result gives a precise characterization of the feature alignment for the two prototypical settings of random features (RF) and neural tangent kernel (NTK) regression. We prove that the memorization of spurious features weakens as the generalization capability increases and, through the analysis of the feature alignment, we unveil the role of the model and of its activation function. Numerical experiments show the predictive power of our theory on standard datasets (MNIST, CIFAR-10).