# Gradient Descent Robustly Learns the Intrinsic Dimension of Data in Training Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2504.08628 · HTML: https://mlsanity.com/paper/2504.08628/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 40 of its 49 references that are in the map.
- First posted on arXiv 11 Apr 2025.

## Builds on (most cited first, 30 of 40)

- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1412.6614/) (2014)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [Towards Understanding Ensemble, Knowledge Distillation and Self-Distillation in Deep Learning](https://mlsanity.com/paper/2012.09816/) (2020)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [Two models of double descent for weak features](https://mlsanity.com/paper/1903.07571/) (2019)
- [Gradient descent aligns the layers of deep linear networks](https://mlsanity.com/paper/1810.02032/) (2018)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [Towards Resolving the Implicit Bias of Gradient Descent for Matrix Factorization: Greedy Low-Rank Learning](https://mlsanity.com/paper/2012.09839/) (2020)
- [Implicit Regularization in Deep Learning May Not Be Explainable by Norms](https://mlsanity.com/paper/2005.06398/) (2020)
- [The Neural Tangent Kernel in High Dimensions: Triple Descent and a Multi-Scale Theory of Generalization](https://mlsanity.com/paper/2008.06786/) (2020)
- [On the Optimal Weighted ℓ2 Regularization in Overparameterized Linear Regression](https://mlsanity.com/paper/2006.05800/) (2020)
- [Vision Transformers provably learn spatial structure](https://mlsanity.com/paper/2210.09221/) (2022)
- [Benign Overfitting in Two-layer Convolutional Neural Networks](https://mlsanity.com/paper/2202.06526/) (2022)
- [On the Implicit Bias in Deep-Learning Algorithms](https://mlsanity.com/paper/2208.12591/) (2022)
- [Gradient Descent on Two-layer Nets: Margin Maximization and Simplicity Bias](https://mlsanity.com/paper/2110.13905/) (2021)
- [Finite-sample Analysis of Interpolating Linear Classifiers in the Overparameterized Regime](https://mlsanity.com/paper/2004.12019/) (2020)
- [Stochastic Gradient Descent on Separable Data: Exact Convergence with a Fixed Learning Rate](https://mlsanity.com/paper/1806.01796/) (2018)
- [Implicit Bias of AdamW: ℓ_∞ Norm Constrained Optimization](https://mlsanity.com/paper/2404.04454/) (2024)
- [Implicit Regularization Towards Rank Minimization in ReLU Networks](https://mlsanity.com/paper/2201.12760/) (2022)
- [Understanding the Generalization of Adam in Learning Neural Networks with Proper Regularization](https://mlsanity.com/paper/2108.11371/) (2021)
- [The Implicit Bias for Adaptive Optimization Algorithms on Homogeneous Neural Networks](https://mlsanity.com/paper/2012.06244/) (2020)
- [Implicit Bias in Leaky ReLU Networks Trained on High-Dimensional Data](https://mlsanity.com/paper/2210.07082/) (2022)
- [Implicit Bias of Large Depth Networks: a Notion of Rank for Nonlinear Functions](https://mlsanity.com/paper/2209.15055/) (2022)

## Abstract (arXiv)

Modern neural networks are usually highly over-parameterized. Behind the wide usage of over-parameterized networks is the belief that, if the data are simple, then the trained network will be automatically equivalent to a simple predictor. Following this intuition, many existing works have studied different notions of "ranks" of neural networks and their relation to the rank of data. In this work, we study the rank of convolutional neural networks (CNNs) trained by gradient descent, with a specific focus on the robustness of the rank to image background noises. Specifically, we point out that, when adding background noises to images, the rank of the CNN trained with gradient descent is affected far less compared with the rank of the data. We support our claim with a theoretical case study, where we consider a particular data model to characterize low-rank clean images with added background noises. We prove that CNNs trained by gradient descent can learn the intrinsic dimension of clean images, despite the presence of relatively large background noises. We also conduct experiments on synthetic and real datasets to further validate our claim.