# The Intrinsic Dimension of Images and Its Impact on Learning

arXiv: https://arxiv.org/abs/2104.08894 · HTML: https://mlsanity.com/paper/2104.08894/

## Key facts

- Cited by 315 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 48 references that are in the map.
- First posted on arXiv 18 Apr 2021.

## Builds on (most cited first, 16 of 16)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [ImageNet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness](https://mlsanity.com/paper/1811.12231/) (2018)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Approximating CNNs with Bag-of-local-Features models works surprisingly well on ImageNet](https://mlsanity.com/paper/1904.00760/) (2019)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Disentangling Adversarial Robustness and Generalization](https://mlsanity.com/paper/1812.00740/) (2018)
- [On the Intrinsic Dimensionality of Image Representations](https://mlsanity.com/paper/1803.09672/) (2018)
- [Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems](https://mlsanity.com/paper/1808.01346/) (2018)
- [Understanding Generalization through Visualizations](https://mlsanity.com/paper/1906.03291/) (2019)
- [LDMNet: Low Dimensional Manifold Regularized Neural Networks](https://mlsanity.com/paper/1711.06246/) (2017)
- [Texture Interpolation for Probing Visual Perception](https://mlsanity.com/paper/2006.03698/) (2020)
- [Geometry-Aware Maximum Likelihood Estimation of Intrinsic Dimension](https://mlsanity.com/paper/1904.06151/) (2019)
- [Minimax Rates for Estimating the Dimension of a Manifold](https://mlsanity.com/paper/1605.01011/) (2016)

## Built on by (most cited first, 30 of 315)

- [StyleGAN-XL: Scaling StyleGAN to Large Diverse Datasets](https://mlsanity.com/paper/2202.00273/) (2022)
- [Guiding a Diffusion Model with a Bad Version of Itself](https://mlsanity.com/paper/2406.02507/) (2024)
- [Diffusion Transformers with Representation Autoencoders](https://mlsanity.com/paper/2510.11690/) (2025)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [A Universal Law of Robustness via Isoperimetry](https://mlsanity.com/paper/2105.12806/) (2021)
- [Score-Based Generative Models Detect Manifolds](https://mlsanity.com/paper/2206.01018/) (2022)
- [Intrinsic Dimension Estimation for Robust Detection of AI-Generated Texts](https://mlsanity.com/paper/2306.04723/) (2023)
- [EQ-VAE: Equivariance Regularized Latent Space for Improved Generative Image Modeling](https://mlsanity.com/paper/2502.09509/) (2025)
- [Better than classical? The subtle art of benchmarking quantum machine learning models](https://mlsanity.com/paper/2403.07059/) (2024)
- [How Deep Neural Networks Learn Compositional Data: The Random Hierarchy Model](https://mlsanity.com/paper/2307.02129/) (2023)
- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [Gradient Guidance for Diffusion Models: An Optimization Perspective](https://mlsanity.com/paper/2404.14743/) (2024)
- [Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models](https://mlsanity.com/paper/2405.14861/) (2024)
- [Emergence of a High-Dimensional Abstraction Phase in Language Transformers](https://mlsanity.com/paper/2405.15471/) (2024)
- [Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions](https://mlsanity.com/paper/2409.02426/) (2024)
- [Neural networks trained with SGD learn distributions of increasing complexity](https://mlsanity.com/paper/2211.11567/) (2022)
- [O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions](https://mlsanity.com/paper/2409.18959/) (2024)
- [Reward-Directed Conditional Diffusion: Provable Distribution Estimation and Reward Improvement](https://mlsanity.com/paper/2307.07055/) (2023)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Characterizing Truthfulness in Large Language Model Generations with Local Intrinsic Dimension](https://mlsanity.com/paper/2402.18048/) (2024)
- [Linear Convergence of Diffusion Models Under the Manifold Hypothesis](https://mlsanity.com/paper/2410.09046/) (2024)
- [Detecting Images Generated by Deep Diffusion Models using their Local Intrinsic Dimensionality](https://mlsanity.com/paper/2307.02347/) (2023)
- [Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality](https://mlsanity.com/paper/2410.18784/) (2024)
- [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024)
- [The Dimpled Manifold Model of Adversarial Examples in Machine Learning](https://mlsanity.com/paper/2106.10151/) (2021)
- [Can Push-forward Generative Models Fit Multimodal Distributions?](https://mlsanity.com/paper/2206.14476/) (2022)
- [On Statistical Rates and Provably Efficient Criteria of Latent Diffusion Transformers (DiTs)](https://mlsanity.com/paper/2407.01079/) (2024)
- [Low-dimensional adaptation of diffusion models: Convergence in total variation](https://mlsanity.com/paper/2501.12982/) (2025)
- [Your diffusion model secretly knows the dimension of the data manifold](https://mlsanity.com/paper/2212.12611/) (2022)
- [Stochastic Self-Guidance for Training-Free Enhancement of Diffusion Models](https://mlsanity.com/paper/2508.12880/) (2025)

## Abstract (arXiv)

It is widely believed that natural image data exhibits low-dimensional structure despite the high dimensionality of conventional pixel representations. This idea underlies a common intuition for the remarkable success of deep learning in computer vision. In this work, we apply dimension estimation tools to popular datasets and investigate the role of low-dimensional structure in deep learning. We find that common natural image datasets indeed have very low intrinsic dimension relative to the high number of pixels in the images. Additionally, we find that low dimensional datasets are easier for neural networks to learn, and models solving these tasks generalize better from training to test data. Along the way, we develop a technique for validating our dimension estimation tools on synthetic data generated by GANs allowing us to actively manipulate the intrinsic dimension by controlling the image generation process. Code for our experiments may be found here https://github.com/ppope/dimensions.