# A Connection Between Score Matching and Local Intrinsic Dimension

arXiv: https://arxiv.org/abs/2510.12975 · HTML: https://mlsanity.com/paper/2510.12975/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 18 of its 32 references that are in the map.
- First posted on arXiv 14 Oct 2025.

## Placed in

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

## Builds on (most cited first, 18 of 18)

- [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)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Scikit-dimension: a Python package for intrinsic dimension estimation](https://mlsanity.com/paper/2109.02596/) (2021)
- [Characterizing Truthfulness in Large Language Model Generations with Local Intrinsic Dimension](https://mlsanity.com/paper/2402.18048/) (2024)
- [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024)
- [LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihood](https://mlsanity.com/paper/2206.14882/) (2022)
- [Adversarial Estimation of Topological Dimension with Harmonic Score Maps](https://mlsanity.com/paper/2312.06869/) (2023)

## Built on by (most cited first, 4 of 4)

- [Local Intrinsic Dimension Unveils Hallucinations in Diffusion Models](https://mlsanity.com/paper/2605.05026/) (2026)
- [Saddle-Free Guidance: Improved On-Manifold Sampling without Labels or Additional Training](https://mlsanity.com/paper/2511.21863/) (2025)
- [FiGuRO: Intrinsic Dimension Estimation for Multi-Modal Data](https://mlsanity.com/paper/2608.10857/) (2026)
- [One Inverse Step is a Convex Program: Bayes-Limit Calibration of Diffusion Inversion](https://mlsanity.com/paper/2608.23094/) (2026)

## Abstract (arXiv)

The local intrinsic dimension (LID) of data is a fundamental quantity in signal processing and learning theory, but quantifying the LID of high-dimensional, complex data has been a historically challenging task. Recent works have discovered that diffusion models capture the LID of data through the spectra of their score estimates and through the rate of change of their density estimates under various noise perturbations. While these methods can accurately quantify LID, they require either many forward passes of the diffusion model or use of gradient computation, limiting their applicability in compute- and memory-constrained scenarios. We show that the LID is a lower bound on the denoising score matching loss, motivating use of the denoising score matching loss as a LID estimator. Moreover, we show that the equivalent implicit score matching loss also approximates LID via the normal dimension and is closely related to a recent LID estimator, FLIPD. Our experiments on a manifold benchmark and with Stable Diffusion 3.5 indicate that the denoising score matching loss is a highly competitive and scalable LID estimator, achieving superior accuracy and memory footprint under increasing problem size and quantization level.