Property Testing of Joint Distributions using Conditional Samples

In this paper, we consider the problem of testing properties of joint distributions under the Conditional Sampling framework. In the standard sampling model, the sample complexity of testing properties of joint distributions is exponential in the dimension, resulting in inefficient algorithms for practical use. While recent results achieve efficient algorithms for product distributions with significantly smaller sample complexity, no efficient algorithm is expected when the marginals are not independent. We initialize the study of conditional sampling in the multidimensional setting. We propose a subcube conditional sampling model where the tester can condition on an (adaptively) chosen subcube of the domain. Due to its simplicity, this model is potentially implementable in many practical applications, particularly when the distribution is a joint distribution over $\Sigma^n$ for some set $\Sigma$. We present algorithms for various fundamental properties of distributions in the subcube-conditioning model and prove that the sample complexity is polynomial in the dimension $n$ (and not exponential as in the traditional model). We present an algorithm for testing identity to a known distribution using $\tilde{\mathcal{O}}(n^{2)$-subcube-conditional} samples, an algorithm for testing identity between two unknown distributions using $\tilde{\mathcal{O}}(n^{5)$-subcube-conditional} samples and an algorithm for testing identity to a product distribution using $tilde{\mathcal{O}}(n^{5)$-subcube-conditional} samples. The central concept of our technique involves an elegant chain rule which can be proved using basic techniques of probability theory yet powerful enough to avoid the curse of dimensionality.