# Sampling and Identity-Testing Without Approximate Tensorization of Entropy

arXiv: https://arxiv.org/abs/2506.23456 · HTML: https://mlsanity.com/paper/2506.23456/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 34 references that are in the map.
- First posted on arXiv 30 Jun 2025.

## Builds on (most cited first, 16 of 16)

- [Optimal Testing for Properties of Distributions](https://mlsanity.com/paper/1507.05952/) (2015)
- [Testing probability distributions underlying aggregated data](https://mlsanity.com/paper/1402.3835/) (2014)
- [Which Distribution Distances are Sublinearly Testable?](https://mlsanity.com/paper/1708.00002/) (2017)
- [Random Restrictions of High-Dimensional Distributions and Uniformity Testing with Subcube Conditioning](https://mlsanity.com/paper/1911.07357/) (2019)
- [Quantum state certification](https://mlsanity.com/paper/1708.06002/) (2017)
- [Learning and Testing Junta Distributions with Subcube Conditioning](https://mlsanity.com/paper/2004.12496/) (2020)
- [Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion](https://mlsanity.com/paper/2011.02075/) (2020)
- [On Mixing of Markov Chains: Coupling, Spectral Independence, and Entropy Factorization](https://mlsanity.com/paper/2103.07459/) (2021)
- [Faster Algorithms for Testing under Conditional Sampling](https://mlsanity.com/paper/1504.04103/) (2015)
- [Quantum chi-squared tomography and mutual information testing](https://mlsanity.com/paper/2305.18519/) (2023)
- [Efficiently learning and sampling multimodal distributions with data-based initialization](https://mlsanity.com/paper/2411.09117/) (2024)
- [On Distribution Testing in the Conditional Sampling Model](https://mlsanity.com/paper/2007.09895/) (2020)
- [Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling](https://mlsanity.com/paper/2207.09102/) (2022)
- [From algorithms to connectivity and back: finding a giant component in random k-SAT](https://mlsanity.com/paper/2207.02841/) (2022)
- [Weak Poincar'e Inequalities, Simulated Annealing, and Sampling from Spherical Spin Glasses](https://mlsanity.com/paper/2411.09075/) (2024)
- [Estimating the Effective Support Size in Constant Query Complexity](https://mlsanity.com/paper/2211.11344/) (2022)

## Built on by (most cited first, 1 of 1)

- [On approximating the f-divergence between two Ising models](https://mlsanity.com/paper/2509.05016/) (2025)

## Abstract (arXiv)

Certain tasks in high-dimensional statistics become easier when the underlying distribution satisfies a local-to-global property called approximate tensorization of entropy (ATE). For example, the Glauber dynamics Markov chain of an ATE distribution mixes fast and can produce approximate samples in a small amount of time, since such a distribution satisfies a modified log-Sobolev inequality. Moreover, identity-testing for an ATE distribution requires few samples if the tester is given coordinate conditional access to the unknown distribution, as shown by Blanca, Chen, vStefankovivc, and Vigoda (COLT 2023). A natural class of distributions that do not satisfy ATE consists of mixtures of (few) distributions that do satisfy ATE. We study the complexity of identity-testing and sampling for these distributions. Our main results are the following: 1. We show fast mixing of Glauber dynamics from a data-based initialization, with optimal sample complexity, for mixtures of distributions satisfying modified log-Sobolev inequalities. This extends work of Huang, Koehler, Lee, Mohanty, Rajaraman, Vuong, and Wu (STOC 2025, COLT 2025) for mixtures of distributions satisfying Poincar'e inequalities. 2. Answering an open question posed by Blanca et al., we give efficient identity-testers for mixtures of ATE distributions in the coordinate-conditional sampling access model. We also give some simplifications and improvements to the original algorithm of Blanca et al.