Complexity of High-Dimensional Identity Testing with Coordinate Conditional Sampling

TL;DR

This work introduces a coordinate-based conditional sampling model for high-dimensional identity testing and shows that approximate tensorization of entropy (AT) for the visible distribution is both sufficient and nearly necessary for efficient testing. The main algorithm reduces the high-dimensional problem to many one-dimensional KL tests on conditional marginals, achieving near-linear in $n$ query complexity under AT, with a matching $\Omega(n/\varepsilon)$ sample lower bound. A computational hardness dichotomy is established for sparse antiferromagnetic Ising models: testing is efficient in the AT regime but becomes NP-hard otherwise (under standard complexity assumptions). The results extend to related models (product distributions, Dobrushin-uniqueness settings) and to Subcube Oracle variants, while also providing lower bounds in TV distance and KL divergence, highlighting the nuanced trade-offs between oracle strength and statistical efficiency. Overall, the paper bridges high-dimensional testing, entropy-decomposition properties, and computational barriers, delivering practical testers under verifiable analytic conditions and outlining compelling open problems in the non-AT regime.

Abstract

We study the identity testing problem for high-dimensional distributions. Given as input an explicit distribution $μ$, an $\varepsilon>0$, and access to sampling oracle(s) for a hidden distribution $π$, the goal in identity testing is to distinguish whether the two distributions $μ$ and $π$ are identical or are at least $\varepsilon$-far apart. When there is only access to full samples from the hidden distribution $π$, it is known that exponentially many samples (in the dimension) may be needed for identity testing, and hence previous works have studied identity testing with additional access to various "conditional" sampling oracles. We consider a significantly weaker conditional sampling oracle, which we call the $\mathsf{Coordinate\ Oracle}$, and provide a computational and statistical characterization of the identity testing problem in this new model. We prove that if an analytic property known as approximate tensorization of entropy holds for an $n$-dimensional visible distribution $μ$, then there is an efficient identity testing algorithm for any hidden distribution $π$ using $\tilde{O}(n/\varepsilon)$ queries to the $\mathsf{Coordinate\ Oracle}$. Approximate tensorization of entropy is a pertinent condition as recent works have established it for a large class of high-dimensional distributions. We also prove a computational phase transition: for a well-studied class of $n$-dimensional distributions, specifically sparse antiferromagnetic Ising models over $\{+1,-1\}^n$, we show that in the regime where approximate tensorization of entropy fails, there is no efficient identity testing algorithm unless $\mathsf{RP}=\mathsf{NP}$. We complement our results with a matching $Ω(n/\varepsilon)$ statistical lower bound for the sample complexity of identity testing in the $\mathsf{Coordinate\ Oracle}$ model.

Theorems & Definitions (89)

• Definition 1.1: Approximate Tensorization of Entropy
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• Remark 2.2