# Sample Complexity Bounds for Estimating Probability Divergences under Invariances

arXiv: https://arxiv.org/abs/2311.02868 · HTML: https://mlsanity.com/paper/2311.02868/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 56 references that are in the map.
- First posted on arXiv 6 Nov 2023.

## Builds on (most cited first, 23 of 23)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Improving GANs Using Optimal Transport](https://mlsanity.com/paper/1803.05573/) (2018)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [How Well Generative Adversarial Networks Learn Distributions](https://mlsanity.com/paper/1811.03179/) (2018)
- [Equivariant Flows: sampling configurations for multi-body systems with symmetric energies](https://mlsanity.com/paper/1910.00753/) (2019)
- [Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces](https://mlsanity.com/paper/1804.10556/) (2018)
- [Sobolev GAN](https://mlsanity.com/paper/1711.04894/) (2017)
- [The Exact Sample Complexity Gain from Invariances for Kernel Regression](https://mlsanity.com/paper/2303.14269/) (2023)
- [Nonparametric Density Estimation under Adversarial Losses](https://mlsanity.com/paper/1805.08836/) (2018)
- [Measuring Generalization with Optimal Transport](https://mlsanity.com/paper/2106.03314/) (2021)
- [Nonparametric Density Estimation & Convergence Rates for GANs under Besov IPM Losses](https://mlsanity.com/paper/1902.03511/) (2019)
- [Scalable Normalizing Flows for Permutation Invariant Densities](https://mlsanity.com/paper/2010.03242/) (2020)
- [Intrinsic Dimension Estimation Using Wasserstein Distances](https://mlsanity.com/paper/2106.04018/) (2021)
- [Group Equivariant Generative Adversarial Networks](https://mlsanity.com/paper/2005.01683/) (2020)
- [Structure-preserving GANs](https://mlsanity.com/paper/2202.01129/) (2022)
- [Sample Complexity of Probability Divergences under Group Symmetry](https://mlsanity.com/paper/2302.01915/) (2023)
- [Estimating Certain Integral Probability Metric (IPM) is as Hard as Estimating under the IPM](https://mlsanity.com/paper/1911.00730/) (2019)
- [(q,p)-Wasserstein GANs: Comparing Ground Metrics for Wasserstein GANs](https://mlsanity.com/paper/1902.03642/) (2019)
- [A Review on Modern Computational Optimal Transport Methods with Applications in Biomedical Research](https://mlsanity.com/paper/2008.02995/) (2020)

## Built on by (most cited first, 12 of 12)

- [SemlaFlow -- Efficient 3D Molecular Generation with Latent Attention and Equivariant Flow Matching](https://mlsanity.com/paper/2406.07266/) (2024)
- [Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetry](https://mlsanity.com/paper/2512.11855/) (2025)
- [Learning with Exact Invariances in Polynomial Time](https://mlsanity.com/paper/2502.19758/) (2025)
- [Robust Generative Learning with Lipschitz-Regularized α-Divergences Allows Minimal Assumptions on Target Distributions](https://mlsanity.com/paper/2405.13962/) (2024)
- [Robustness and Structure Preservation in Flow-Based Generative Models via Wasserstein Path-Space Divergences](https://mlsanity.com/paper/2410.01244/) (2024)
- [Adaptive Symmetry Discovery for Dynamical System Identification](https://mlsanity.com/paper/2608.08091/) (2026)
- [Efficient Learning and Symmetry Discovery under Exact Invariances](https://mlsanity.com/paper/2609.07031/) (2026)
- [Error Analysis of Bayesian Inverse Problems with Generative Priors](https://mlsanity.com/paper/2601.17374/) (2026)
- [Statistical Learning Guarantees for Group-Invariant Barron Functions](https://mlsanity.com/paper/2509.23474/) (2025)
- [GEPC: Group-Equivariant Posterior Consistency for Out-of-Distribution Detection in Diffusion Models](https://mlsanity.com/paper/2602.00191/) (2026)
- [Sparse Data Augmentation for Optimization with Provable Guarantees](https://mlsanity.com/paper/2609.08133/) (2026)
- [Dimension-Free Scaling Laws for Size Generalization: An Exponential Separation](https://mlsanity.com/paper/2610.19279/) (2026)

## Abstract (arXiv)

Group-invariant probability distributions appear in many data-generative models in machine learning, such as graphs, point clouds, and images. In practice, one often needs to estimate divergences between such distributions. In this work, we study how the inherent invariances, with respect to any smooth action of a Lie group on a manifold, improve sample complexity when estimating the 1-Wasserstein distance, the Sobolev Integral Probability Metrics (Sobolev IPMs), the Maximum Mean Discrepancy (MMD), and also the complexity of the density estimation problem (in the L2 and L^∞ distance). Our results indicate a two-fold gain: (1) reducing the sample complexity by a multiplicative factor corresponding to the group size (for finite groups) or the normalized volume of the quotient space (for groups of positive dimension); (2) improving the exponent in the convergence rate (for groups of positive dimension). These results are completely new for groups of positive dimension and extend recent bounds for finite group actions.