# Sparse Priors for Efficient Distribution Learning

arXiv: https://arxiv.org/abs/2609.20883 · HTML: https://mlsanity.com/paper/2609.20883/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 35 references that are in the map.
- First posted on arXiv 16 Sep 2026.

## Builds on (most cited first, 15 of 15)

- [Scaling Laws for Autoregressive Generative Modeling](https://mlsanity.com/paper/2010.14701/) (2020)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [How Well Generative Adversarial Networks Learn Distributions](https://mlsanity.com/paper/1811.03179/) (2018)
- [Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory](https://mlsanity.com/paper/2403.11968/) (2024)
- [On the Intrinsic Dimensionality of Image Representations](https://mlsanity.com/paper/1803.09672/) (2018)
- [Nonparametric Density Estimation under Adversarial Losses](https://mlsanity.com/paper/1805.08836/) (2018)
- [We Should Chart an Atlas of All the World's Models](https://mlsanity.com/paper/2503.10633/) (2025)
- [Distribution-Free Distribution Regression](https://mlsanity.com/paper/1302.0082/) (2013)
- [Provably tuning the ElasticNet across instances](https://mlsanity.com/paper/2207.10199/) (2022)
- [Algorithm- and Data-Dependent Generalization Bounds for Diffusion Models](https://mlsanity.com/paper/2506.03849/) (2025)
- [Distribution-dependent Generalization Bounds for Tuning Linear Regression Across Tasks](https://mlsanity.com/paper/2507.05084/) (2025)
- [Generalization Guarantees on Data-Driven Tuning of Gradient Descent with Langevin Updates](https://mlsanity.com/paper/2604.13130/) (2026)

## Abstract (arXiv)

Despite the widespread use and success of generative AI techniques today, theoretical guarantees on learning a distribution supported in d dimensions from n samples degrade as O(n-1/Theta(d)), though shown to be minimax optimal. We hypothesize that present bounds are too pessimistic because smoothness assumptions are not enough to capture the structure of distributions that often appear in real applications. Consequently, we introduce the class of sparse priors and define the "Sparse Dimension" as a measure of sparsity of a prior over the space of all distributions. We show that distribution learning under a k-sparse prior achieves a Bayesian risk lower bound of Omega(√k/n) under common distance metrics, and show a matching (up to logarithmic terms asymptotically in n,k) upper bound for the TV distance under mild additional assumptions. We show the statistical equivalence of distribution learning and learning to sample in the Bayesian setting so that our results apply to learning to sample as well. While k can still depend on the dimension d, or a notion of intrinsic dimension, our results show that learning under an appropriate prior overcomes the curse of dimensionality with respect to the dependence on n.