# Intrinsic Dimension Estimation Using Wasserstein Distances

arXiv: https://arxiv.org/abs/2106.04018 · HTML: https://mlsanity.com/paper/2106.04018/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 77 references that are in the map.
- First posted on arXiv 8 Jun 2021.

## Builds on (most cited first, 9 of 9)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [LOGAN: Latent Optimisation for Generative Adversarial Networks](https://mlsanity.com/paper/1912.00953/) (2019)
- [Statistical guarantees for generative models without domination](https://mlsanity.com/paper/2010.09237/) (2020)
- [Nonparametric Density Estimation & Convergence Rates for GANs under Besov IPM Losses](https://mlsanity.com/paper/1902.03511/) (2019)
- [Adaptive Metric Dimensionality Reduction](https://mlsanity.com/paper/1302.2752/) (2013)
- [Fast Mixing of Multi-Scale Langevin Dynamics under the Manifold Hypothesis](https://mlsanity.com/paper/2006.11166/) (2020)
- [Minimax Rates for Estimating the Dimension of a Manifold](https://mlsanity.com/paper/1605.01011/) (2016)
- [Functions with average smoothness: structure, algorithms, and learning](https://mlsanity.com/paper/2007.06283/) (2020)
- [Adaptive Nonparametric Clustering](https://mlsanity.com/paper/1709.09102/) (2017)

## Built on by (most cited first, 20 of 20)

- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality](https://mlsanity.com/paper/2410.18784/) (2024)
- [Discovering Conservation Laws using Optimal Transport and Manifold Learning](https://mlsanity.com/paper/2208.14995/) (2022)
- [Sample Complexity Bounds for Estimating Probability Divergences under Invariances](https://mlsanity.com/paper/2311.02868/) (2023)
- [Relating Regularization and Generalization through the Intrinsic Dimension of Activations](https://mlsanity.com/paper/2211.13239/) (2022)
- [Data-Copying in Generative Models: A Formal Framework](https://mlsanity.com/paper/2302.13181/) (2023)
- [CA-PCA: Manifold Dimension Estimation, Adapted for Curvature](https://mlsanity.com/paper/2309.13478/) (2023)
- [Deep Generative Models: Complexity, Dimensionality, and Approximation](https://mlsanity.com/paper/2504.00820/) (2025)
- [Scale adaptive and robust intrinsic dimension estimation via optimal neighbourhood identification](https://mlsanity.com/paper/2405.15132/) (2024)
- [Generative adversarial learning with optimal input dimension and its adaptive generator architecture](https://mlsanity.com/paper/2405.03723/) (2024)
- [Blessing of Dimensionality for Approximating Sobolev Classes on Manifolds](https://mlsanity.com/paper/2408.06996/) (2024)
- [Geometric Collaborative Filtering with Convergence](https://mlsanity.com/paper/2410.03064/) (2024)
- [Rate-Optimal Noise Annealing in Semi-Dual Neural Optimal Transport: Tangential Identifiability, Off-Manifold Ambiguity, and Guaranteed Recovery](https://mlsanity.com/paper/2602.04110/) (2026)
- [Manifold Regularization for Memory-Efficient Training of Deep Neural Networks](https://mlsanity.com/paper/2305.17119/) (2023)
- [On consistent estimation of dimension values](https://mlsanity.com/paper/2412.13898/) (2024)
- [A Survey and Comparative Evaluation of Intrinsic Dimension Estimators under the Manifold Hypothesis](https://mlsanity.com/paper/2509.15517/) (2025)
- [HomID : Benchmarking Intrinsic Dimension Estimators on Homogenous Manifolds with Anisotropic Embeddings](https://mlsanity.com/paper/2510.01335/) (2025)
- [Manifold Dimension Estimation via Local Graph Structure](https://mlsanity.com/paper/2510.15141/) (2025)
- [Geometry-Aware Optimal Transport: Fast Intrinsic Dimension and Wasserstein Distance Estimation](https://mlsanity.com/paper/2602.04335/) (2026)
- [A Unified Framework for In-Context Learning with Causal and Masked Language Models](https://mlsanity.com/paper/2607.04081/) (2026)

## Abstract (arXiv)

It has long been thought that high-dimensional data encountered in many practical machine learning tasks have low-dimensional structure, i.e., the manifold hypothesis holds. A natural question, thus, is to estimate the intrinsic dimension of a given population distribution from a finite sample. We introduce a new estimator of the intrinsic dimension and provide finite sample, non-asymptotic guarantees. We then apply our techniques to get new sample complexity bounds for Generative Adversarial Networks (GANs) depending only on the intrinsic dimension of the data.