# On Convolutions, Intrinsic Dimension, and Diffusion Models arXiv: https://arxiv.org/abs/2506.20705 ยท HTML: https://mlsanity.com/paper/2506.20705/ ## Key facts - Cited by 3 papers in the map (data through 14 Oct 2026). - Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. - Builds on 46 of its 70 references that are in the map. - First posted on arXiv 25 Jun 2025. ## Placed in - Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/) ## Builds on (most cited first, 30 of 46) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [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) - [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022) - [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022) - [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016) - [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022) - [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023) - [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019) - [Reliable Fidelity and Diversity Metrics for Generative Models](https://mlsanity.com/paper/2002.09797/) (2020) - [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021) - [Understanding Diffusion Objectives as the ELBO with Simple Data Augmentation](https://mlsanity.com/paper/2303.00848/) (2023) - [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) - [Diagnosing and Enhancing VAE Models](https://mlsanity.com/paper/1903.05789/) (2019) - [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023) - [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023) - [Score-Based Generative Models Detect Manifolds](https://mlsanity.com/paper/2206.01018/) (2022) - [Flows for simultaneous manifold learning and density estimation](https://mlsanity.com/paper/2003.13913/) (2020) - [Intrinsic Dimension Estimation for Robust Detection of AI-Generated Texts](https://mlsanity.com/paper/2306.04723/) (2023) - [SoftFlow: Probabilistic Framework for Normalizing Flow on Manifolds](https://mlsanity.com/paper/2006.04604/) (2020) - [Generalized Energy Based Models](https://mlsanity.com/paper/2003.05033/) (2020) - [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) - [Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions](https://mlsanity.com/paper/2409.02426/) (2024) - [Rectangular Flows for Manifold Learning](https://mlsanity.com/paper/2106.01413/) (2021) ## Built on by (most cited first, 3 of 3) - [A Deep Generative Approach to Stratified Learning](https://mlsanity.com/paper/2604.10650/) (2026) - [Local Hessian Spectral Filtering for Robust Intrinsic Dimension Estimation](https://mlsanity.com/paper/2605.01221/) (2026) - [Conf-Gen: Conformal Uncertainty Quantification for Generative Models](https://mlsanity.com/paper/2605.28920/) (2026) ## Abstract (arXiv) The manifold hypothesis asserts that data of interest in high-dimensional ambient spaces, such as image data, lies on unknown low-dimensional submanifolds. Diffusion models (DMs) -- which operate by convolving data with progressively larger amounts of Gaussian noise and then learning to revert this process -- have risen to prominence as the most performant generative models, and are known to be able to learn distributions with low-dimensional support. For a given datum in one of these submanifolds, we should thus intuitively expect DMs to have implicitly learned its corresponding local intrinsic dimension (LID), i.e. the dimension of the submanifold it belongs to. Kamkari et al. (2024b) recently showed that this is indeed the case by linking this LID to the rate of change of the log marginal densities of the DM with respect to the amount of added noise, resulting in an LID estimator known as FLIPD. LID estimators such as FLIPD have a plethora of uses, among others they quantify the complexity of a given datum, and can be used to detect outliers, adversarial examples and AI-generated text. FLIPD achieves state-of-the-art performance at LID estimation, yet its theoretical underpinnings are incomplete since Kamkari et al. (2024b) only proved its correctness under the highly unrealistic assumption of affine submanifolds. In this work we bridge this gap by formally proving the correctness of FLIPD under realistic assumptions. Additionally, we show that an analogous result holds when Gaussian convolutions are replaced with uniform ones, and discuss the relevance of this result.