# LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihood

arXiv: https://arxiv.org/abs/2206.14882 · HTML: https://mlsanity.com/paper/2206.14882/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 45 references that are in the map.
- First posted on arXiv 29 Jun 2022.

## Builds on (most cited first, 20 of 20)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Measuring the Intrinsic Dimension of Objective Landscapes](https://mlsanity.com/paper/1804.08838/) (2018)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [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)
- [Flows for simultaneous manifold learning and density estimation](https://mlsanity.com/paper/2003.13913/) (2020)
- [Scikit-dimension: a Python package for intrinsic dimension estimation](https://mlsanity.com/paper/2109.02596/) (2021)
- [Rectangular Flows for Manifold Learning](https://mlsanity.com/paper/2106.01413/) (2021)
- [Tractable Density Estimation on Learned Manifolds with Conformal Embedding Flows](https://mlsanity.com/paper/2106.05275/) (2021)
- [On the Latent Space of Wasserstein Auto-Encoders](https://mlsanity.com/paper/1802.03761/) (2018)
- [Estimating the effective dimension of large biological datasets using Fisher separability analysis](https://mlsanity.com/paper/1901.06328/) (2019)
- [Unsupervised representation learning using convolutional and stacked auto-encoders: a domain and cross-domain feature space analysis](https://mlsanity.com/paper/1811.00473/) (2018)

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

- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024)
- [A Geometric Framework for Understanding Memorization in Generative Models](https://mlsanity.com/paper/2411.00113/) (2024)
- [On gauge freedom, conservativity and intrinsic dimensionality estimation in diffusion models](https://mlsanity.com/paper/2402.03845/) (2024)
- [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)
- [Learning normalized image densities via dual score matching](https://mlsanity.com/paper/2506.05310/) (2025)
- [Relating Regularization and Generalization through the Intrinsic Dimension of Activations](https://mlsanity.com/paper/2211.13239/) (2022)
- [Denoising Deep Generative Models](https://mlsanity.com/paper/2212.01265/) (2022)
- [LeapFactual: Reliable Visual Counterfactual Explanation Using Conditional Flow Matching](https://mlsanity.com/paper/2510.14623/) (2025)
- [One-Line-of-Code Data Mollification Improves Optimization of Likelihood-based Generative Models](https://mlsanity.com/paper/2305.18900/) (2023)
- [Adversarial Estimation of Topological Dimension with Harmonic Score Maps](https://mlsanity.com/paper/2312.06869/) (2023)
- [A Connection Between Score Matching and Local Intrinsic Dimension](https://mlsanity.com/paper/2510.12975/) (2025)
- [Impact of dataset size and long-term ECoG-based BCI usage on deep learning decoders performance](https://mlsanity.com/paper/2209.03789/) (2022)
- [Score-based Pullback Riemannian Geometry: Extracting the Data Manifold Geometry using Anisotropic Flows](https://mlsanity.com/paper/2410.01950/) (2024)
- [On Convolutions, Intrinsic Dimension, and Diffusion Models](https://mlsanity.com/paper/2506.20705/) (2025)
- [IdEst: Assessing Self-Supervised Learning Representations via Intrinsic Dimension](https://mlsanity.com/paper/2606.03338/) (2026)
- [A survey of manifold learning and its applications for multimedia](https://mlsanity.com/paper/2310.12986/) (2023)
- [Dimensionality-Aware Outlier Detection: Theoretical and Experimental Analysis](https://mlsanity.com/paper/2401.05453/) (2024)
- [A Wiener Process Perspective on Local Intrinsic Dimension Estimation Methods](https://mlsanity.com/paper/2406.17125/) (2024)
- [Geometry-Guided Adversarial Prompt Detection via Curvature and Local Intrinsic Dimension](https://mlsanity.com/paper/2503.03502/) (2025)
- [Estimating Dataset Dimension via Singular Metrics under the Manifold Hypothesis: Application to Inverse Problems](https://mlsanity.com/paper/2507.07291/) (2025)
- [A Universal Nearest-Neighbor Estimator for Intrinsic Dimensionality](https://mlsanity.com/paper/2603.10493/) (2026)
- [Local Intrinsic Dimension Unveils Hallucinations in Diffusion Models](https://mlsanity.com/paper/2605.05026/) (2026)
- [SoftSAE: Dynamic Top-K Selection for Adaptive Sparse Autoencoders](https://mlsanity.com/paper/2605.06610/) (2026)
- [Style-CCL: Content-Preserving Style Transfer via Curriculum Continual Learning](https://mlsanity.com/paper/2606.14746/) (2026)
- [HomID : Benchmarking Intrinsic Dimension Estimators on Homogenous Manifolds with Anisotropic Embeddings](https://mlsanity.com/paper/2510.01335/) (2025)
- [On the Use of Bagging for Local Intrinsic Dimensionality Estimation](https://mlsanity.com/paper/2603.24384/) (2026)
- [One Inverse Step is a Convex Program: Bayes-Limit Calibration of Diffusion Inversion](https://mlsanity.com/paper/2608.23094/) (2026)
- [Minimax Lower Bound for Estimating Diffusion-based Local Intrinsic Dimension](https://mlsanity.com/paper/2609.04822/) (2026)

## Abstract (arXiv)

Most of the existing methods for estimating the local intrinsic dimension of a data distribution do not scale well to high-dimensional data. Many of them rely on a non-parametric nearest neighbors approach which suffers from the curse of dimensionality. We attempt to address that challenge by proposing a novel approach to the problem: Local Intrinsic Dimension estimation using approximate Likelihood (LIDL). Our method relies on an arbitrary density estimation method as its subroutine and hence tries to sidestep the dimensionality challenge by making use of the recent progress in parametric neural methods for likelihood estimation. We carefully investigate the empirical properties of the proposed method, compare them with our theoretical predictions, and show that LIDL yields competitive results on the standard benchmarks for this problem and that it scales to thousands of dimensions. What is more, we anticipate this approach to improve further with the continuing advances in the density estimation literature.