# A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models

arXiv: https://arxiv.org/abs/2406.03537 · HTML: https://mlsanity.com/paper/2406.03537/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 42 of its 71 references that are in the map.
- First posted on arXiv 5 Jun 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 42)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [LAION-5B: An open large-scale dataset for training next generation image-text models](https://mlsanity.com/paper/2210.08402/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Characterizing Adversarial Subspaces Using Local Intrinsic Dimensionality](https://mlsanity.com/paper/1801.02613/) (2018)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Why Normalizing Flows Fail to Detect Out-of-Distribution Data](https://mlsanity.com/paper/2006.08545/) (2020)
- [Score-Based Generative Models Detect Manifolds](https://mlsanity.com/paper/2206.01018/) (2022)
- [The geometry of hidden representations of large transformer models](https://mlsanity.com/paper/2302.00294/) (2023)
- [Intrinsic Dimension Estimation for Robust Detection of AI-Generated Texts](https://mlsanity.com/paper/2306.04723/) (2023)
- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [Scikit-dimension: a Python package for intrinsic dimension estimation](https://mlsanity.com/paper/2109.02596/) (2021)
- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [Your diffusion model secretly knows the dimension of the data manifold](https://mlsanity.com/paper/2212.12611/) (2022)
- [LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihood](https://mlsanity.com/paper/2206.14882/) (2022)

## Built on by (most cited first, 30 of 42)

- [Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions](https://mlsanity.com/paper/2409.02426/) (2024)
- [A Geometric Framework for Understanding Memorization in Generative Models](https://mlsanity.com/paper/2411.00113/) (2024)
- [Losing dimensions: Geometric memorization in generative diffusion](https://mlsanity.com/paper/2410.08727/) (2024)
- [Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling](https://mlsanity.com/paper/2504.10612/) (2025)
- [Diffusion Models as Cartoonists: The Curious Case of High Density Regions](https://mlsanity.com/paper/2411.01293/) (2024)
- [Understanding and Mitigating Memorization in Generative Models via Sharpness of Probability Landscapes](https://mlsanity.com/paper/2412.04140/) (2024)
- [Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptive](https://mlsanity.com/paper/2510.02305/) (2025)
- [Learning normalized image densities via dual score matching](https://mlsanity.com/paper/2506.05310/) (2025)
- [On the Interpolation Effect of Score Smoothing in Diffusion Models](https://mlsanity.com/paper/2502.19499/) (2025)
- [Provable Maximum Entropy Manifold Exploration via Diffusion Models](https://mlsanity.com/paper/2506.15385/) (2025)
- [Decentralized Diffusion Models](https://mlsanity.com/paper/2501.05450/) (2025)
- [Generalization of Diffusion Models Arises with a Balanced Representation Space](https://mlsanity.com/paper/2512.20963/) (2025)
- [Landing with the Score: Riemannian Optimization through Denoising](https://mlsanity.com/paper/2509.23357/) (2025)
- [Devil is in the Details: Density Guidance for Detail-Aware Generation with Flow Models](https://mlsanity.com/paper/2502.05807/) (2025)
- [Image Interpolation with Score-based Riemannian Metrics of Diffusion Models](https://mlsanity.com/paper/2504.20288/) (2025)
- [A Connection Between Score Matching and Local Intrinsic Dimension](https://mlsanity.com/paper/2510.12975/) (2025)
- [Iso-Riemannian Optimization on Learned Data Manifolds](https://mlsanity.com/paper/2510.21033/) (2025)
- [Sparse Autoencoders, Again?](https://mlsanity.com/paper/2506.04859/) (2025)
- [On Convolutions, Intrinsic Dimension, and Diffusion Models](https://mlsanity.com/paper/2506.20705/) (2025)
- [Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models](https://mlsanity.com/paper/2510.05509/) (2025)
- [Diffusion Models Are Statistically Optimal for Learning Low-Dimensional Multi-Modal Distributions](https://mlsanity.com/paper/2605.30153/) (2026)
- [DRESS: Disentangled Representation-based Self-Supervised Meta-Learning for Diverse Tasks](https://mlsanity.com/paper/2503.09679/) (2025)
- [ELROND: Exploring and decomposing intrinsic capabilities of diffusion models](https://mlsanity.com/paper/2602.10216/) (2026)
- [A Wiener Process Perspective on Local Intrinsic Dimension Estimation Methods](https://mlsanity.com/paper/2406.17125/) (2024)
- [Shallow Diffuse: Robust and Invisible Watermarking through Low-Dimensional Subspaces in Diffusion Models](https://mlsanity.com/paper/2410.21088/) (2024)
- [Estimating Dataset Dimension via Singular Metrics under the Manifold Hypothesis: Application to Inverse Problems](https://mlsanity.com/paper/2507.07291/) (2025)
- [Multi-Subspace Multi-Modal Modeling for Diffusion Models: Estimation, Convergence and Mixture of Experts](https://mlsanity.com/paper/2601.01475/) (2026)
- [Reducing Memorisation in Generative Models via Riemannian Bayesian Inference](https://mlsanity.com/paper/2602.00199/) (2026)
- [Local Hessian Spectral Filtering for Robust Intrinsic Dimension Estimation](https://mlsanity.com/paper/2605.01221/) (2026)
- [Local Intrinsic Dimension Unveils Hallucinations in Diffusion Models](https://mlsanity.com/paper/2605.05026/) (2026)

## Abstract (arXiv)

High-dimensional data commonly lies on low-dimensional submanifolds, and estimating the local intrinsic dimension (LID) of a datum -- i.e. the dimension of the submanifold it belongs to -- is a longstanding problem. LID can be understood as the number of local factors of variation: the more factors of variation a datum has, the more complex it tends to be. Estimating this quantity has proven useful in contexts ranging from generalization in neural networks to detection of out-of-distribution data, adversarial examples, and AI-generated text. The recent successes of deep generative models present an opportunity to leverage them for LID estimation, but current methods based on generative models produce inaccurate estimates, require more than a single pre-trained model, are computationally intensive, or do not exploit the best available deep generative models: diffusion models (DMs). In this work, we show that the Fokker-Planck equation associated with a DM can provide an LID estimator which addresses the aforementioned deficiencies. Our estimator, called FLIPD, is easy to implement and compatible with all popular DMs. Applying FLIPD to synthetic LID estimation benchmarks, we find that DMs implemented as fully-connected networks are highly effective LID estimators that outperform existing baselines. We also apply FLIPD to natural images where the true LID is unknown. Despite being sensitive to the choice of network architecture, FLIPD estimates remain a useful measure of relative complexity; compared to competing estimators, FLIPD exhibits a consistently higher correlation with image PNG compression rate and better aligns with qualitative assessments of complexity. Notably, FLIPD is orders of magnitude faster than other LID estimators, and the first to be tractable at the scale of Stable Diffusion.