# The Low-Rank Simplicity Bias in Deep Networks

arXiv: https://arxiv.org/abs/2103.10427 · HTML: https://mlsanity.com/paper/2103.10427/

## Key facts

- Cited by 154 papers in the map (data through 14 Oct 2026).
- Builds on 61 of its 95 references that are in the map.
- First posted on arXiv 18 Mar 2021.

## Builds on (most cited first, 30 of 61)

- [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)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Aggregated Residual Transformations for Deep Neural Networks](https://mlsanity.com/paper/1611.05431/) (2016)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Neural Architecture Search with Reinforcement Learning](https://mlsanity.com/paper/1611.01578/) (2016)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Deep Learning Scaling is Predictable, Empirically](https://mlsanity.com/paper/1712.00409/) (2017)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [The Pitfalls of Simplicity Bias in Neural Networks](https://mlsanity.com/paper/2006.07710/) (2020)
- [In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1412.6614/) (2014)
- [Skip-Thought Vectors](https://mlsanity.com/paper/1506.06726/) (2015)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)

## Built on by (most cited first, 30 of 154)

- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [The Platonic Representation Hypothesis](https://mlsanity.com/paper/2405.07987/) (2024)
- [When More is Less: Understanding Chain-of-Thought Length in LLMs](https://mlsanity.com/paper/2502.07266/) (2025)
- [Can contrastive learning avoid shortcut solutions?](https://mlsanity.com/paper/2106.11230/) (2021)
- [KV Cache is 1 Bit Per Channel: Efficient Large Language Model Inference with Coupled Quantization](https://mlsanity.com/paper/2405.03917/) (2024)
- [The No Free Lunch Theorem, Kolmogorov Complexity, and the Role of Inductive Biases in Machine Learning](https://mlsanity.com/paper/2304.05366/) (2023)
- [The Difficulty of Passive Learning in Deep Reinforcement Learning](https://mlsanity.com/paper/2110.14020/) (2021)
- [Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks](https://mlsanity.com/paper/2310.09528/) (2023)
- [Neural Redshift: Random Networks are not Random Functions](https://mlsanity.com/paper/2403.02241/) (2024)
- [PLASTIC: Improving Input and Label Plasticity for Sample Efficient Reinforcement Learning](https://mlsanity.com/paper/2306.10711/) (2023)
- [Stochastic Collapse: How Gradient Noise Attracts SGD Dynamics Towards Simpler Subnetworks](https://mlsanity.com/paper/2306.04251/) (2023)
- [Sharpness-Aware Minimization Leads to Low-Rank Features](https://mlsanity.com/paper/2305.16292/) (2023)
- [Unsupervised 3D Perception with 2D Vision-Language Distillation for Autonomous Driving](https://mlsanity.com/paper/2309.14491/) (2023)
- [The Tunnel Effect: Building Data Representations in Deep Neural Networks](https://mlsanity.com/paper/2305.19753/) (2023)
- [Training invariances and the low-rank phenomenon: beyond linear networks](https://mlsanity.com/paper/2201.11968/) (2022)
- [Approaching Deep Learning through the Spectral Dynamics of Weights](https://mlsanity.com/paper/2408.11804/) (2024)
- [Deep neural networks have an inbuilt Occam's razor](https://mlsanity.com/paper/2304.06670/) (2023)
- [CoLA: Compute-Efficient Pre-Training of LLMs via Low-Rank Activation](https://mlsanity.com/paper/2502.10940/) (2025)
- [A Closer Look at Multimodal Representation Collapse](https://mlsanity.com/paper/2505.22483/) (2025)
- [No Representation, No Trust: Connecting Representation, Collapse, and Trust Issues in PPO](https://mlsanity.com/paper/2405.00662/) (2024)
- [Simplicity Bias in 1-Hidden Layer Neural Networks](https://mlsanity.com/paper/2302.00457/) (2023)
- [Compressible Dynamics in Deep Overparameterized Low-Rank Learning & Adaptation](https://mlsanity.com/paper/2406.04112/) (2024)
- [An Analytical Theory of Spectral Bias in the Learning Dynamics of Diffusion Models](https://mlsanity.com/paper/2503.03206/) (2025)
- [Mechanism of feature learning in deep fully connected networks and kernel machines that recursively learn features](https://mlsanity.com/paper/2212.13881/) (2022)
- [An Empirical Study of Implicit Regularization in Deep Offline RL](https://mlsanity.com/paper/2207.02099/) (2022)
- [Plasticity Loss in Deep Reinforcement Learning: A Survey](https://mlsanity.com/paper/2411.04832/) (2024)
- [Learned feature representations are biased by complexity, learning order, position, and more](https://mlsanity.com/paper/2405.05847/) (2024)
- [The Law of Parsimony in Gradient Descent for Learning Deep Linear Networks](https://mlsanity.com/paper/2306.01154/) (2023)
- [Dynamics Generalisation in Reinforcement Learning via Adaptive Context-Aware Policies](https://mlsanity.com/paper/2310.16686/) (2023)
- [Block-Recurrent Dynamics in Vision Transformers](https://mlsanity.com/paper/2512.19941/) (2025)

## Abstract (arXiv)

Modern deep neural networks are highly over-parameterized compared to the data on which they are trained, yet they often generalize remarkably well. A flurry of recent work has asked: why do deep networks not overfit to their training data? In this work, we make a series of empirical observations that investigate and extend the hypothesis that deeper networks are inductively biased to find solutions with lower effective rank embeddings. We conjecture that this bias exists because the volume of functions that maps to low effective rank embedding increases with depth. We show empirically that our claim holds true on finite width linear and non-linear models on practical learning paradigms and show that on natural data, these are often the solutions that generalize well. We then show that the simplicity bias exists at both initialization and after training and is resilient to hyper-parameters and learning methods. We further demonstrate how linear over-parameterization of deep non-linear models can be used to induce low-rank bias, improving generalization performance on CIFAR and ImageNet without changing the modeling capacity.