# Sample Complexity of Probability Divergences under Group Symmetry

arXiv: https://arxiv.org/abs/2302.01915 · HTML: https://mlsanity.com/paper/2302.01915/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 72 references that are in the map.
- First posted on arXiv 3 Feb 2023.

## Builds on (most cited first, 27 of 27)

- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Differentiable Augmentation for Data-Efficient GAN Training](https://mlsanity.com/paper/2006.10738/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [Graph Normalizing Flows](https://mlsanity.com/paper/1905.13177/) (2019)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Equivariant Flows: sampling configurations for multi-body systems with symmetric energies](https://mlsanity.com/paper/1910.00753/) (2019)
- [Efficient Estimation of Mutual Information for Strongly Dependent Variables](https://mlsanity.com/paper/1411.2003/) (2014)
- [Equitability, mutual information, and the maximal information coefficient](https://mlsanity.com/paper/1301.7745/) (2013)
- [Generalization Error of Invariant Classifiers](https://mlsanity.com/paper/1610.04574/) (2016)
- [Equivariant Hamiltonian Flows](https://mlsanity.com/paper/1909.13739/) (2019)
- [Large-Scale Kernel Methods for Independence Testing](https://mlsanity.com/paper/1606.07892/) (2016)
- [Neural Estimation of Statistical Divergences](https://mlsanity.com/paper/2110.03652/) (2021)
- [Scalable Normalizing Flows for Permutation Invariant Densities](https://mlsanity.com/paper/2010.03242/) (2020)
- [Variational Representations and Neural Network Estimation of R'enyi Divergences](https://mlsanity.com/paper/2007.03814/) (2020)
- [Group Equivariant Generative Adversarial Networks](https://mlsanity.com/paper/2005.01683/) (2020)
- [Structure-preserving GANs](https://mlsanity.com/paper/2202.01129/) (2022)
- [Optimizing Variational Representations of Divergences and Accelerating their Statistical Estimation](https://mlsanity.com/paper/2006.08781/) (2020)
- [Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications](https://mlsanity.com/paper/2101.04039/) (2021)
- [Kernel Two-Sample Tests for Manifold Data](https://mlsanity.com/paper/2105.03425/) (2021)
- [Function-space regularized R'enyi divergences](https://mlsanity.com/paper/2210.04974/) (2022)

## Built on by (most cited first, 14 of 14)

- [Sample Complexity Bounds for Estimating Probability Divergences under Invariances](https://mlsanity.com/paper/2311.02868/) (2023)
- [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)
- [Statistical Guarantees of Group-Invariant GANs](https://mlsanity.com/paper/2305.13517/) (2023)
- [Robust Generative Learning with Lipschitz-Regularized α-Divergences Allows Minimal Assumptions on Target Distributions](https://mlsanity.com/paper/2405.13962/) (2024)
- [Nonlinear denoising score matching for enhanced learning of structured distributions](https://mlsanity.com/paper/2405.15625/) (2024)
- [Robustness and Structure Preservation in Flow-Based Generative Models via Wasserstein Path-Space Divergences](https://mlsanity.com/paper/2410.01244/) (2024)
- [Efficient Learning and Symmetry Discovery under Exact Invariances](https://mlsanity.com/paper/2609.07031/) (2026)
- [Statistical Error Bounds for GANs with Nonlinear Objective Functionals](https://mlsanity.com/paper/2406.16834/) (2024)
- [Robust training of implicit generative models for multivariate and heavy-tailed distributions with an invariant statistical loss](https://mlsanity.com/paper/2410.22381/) (2024)
- [Bias Detection via Maximum Subgroup Discrepancy](https://mlsanity.com/paper/2502.02221/) (2025)
- [Statistical Learning Guarantees for Group-Invariant Barron Functions](https://mlsanity.com/paper/2509.23474/) (2025)
- [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)

We rigorously quantify the improvement in the sample complexity of variational divergence estimations for group-invariant distributions. In the cases of the Wasserstein-1 metric and the Lipschitz-regularized α-divergences, the reduction of sample complexity is proportional to the group size if the group is finite. In addition to the published version at ICML 2023, our proof indeed has included the case when the group is infinite such as compact Lie groups, the convergence rate can be further improved and depends on the intrinsic dimension of the fundamental domain characterized by the scaling of its covering number. Our approach is different from that in [Tahmasebi & Jegelka, ICML 2024] and our work also applies to asymmetric divergences, such as the Lipschitz-regularized α-divergences. For the maximum mean discrepancy (MMD), the improvement of sample complexity is more nuanced, as it depends on not only the group size but also the choice of kernel. Numerical simulations verify our theories.