Learning and Testing Junta Distributions with Subcube Conditioning

We study the problems of learning and testing junta distributions on {-1, 1}ⁿ with respect to the uniform distribution, where a distribution p is a k-junta if its probability mass function p(x) depends on a subset of at most k variables. The main contribution is an algorithm for finding relevant coordinates in a k-junta distribution with subcube conditioning Bhattacharyya and Chakraborty (2018); Canonne et al. (2019). We give two applications: • An algorithm for learning k-junta distributions with Õ(k/ε²) log n + O(2^k/ε²) subcube conditioning queries, and • An algorithm for testing k-junta distributions with Õ((k + √n)/ε²) subcube conditioning queries. All our algorithms are optimal up to poly-logarithmic factors. Our results show that subcube conditioning, as a natural model for accessing high-dimensional distributions, enables significant savings in learning and testing junta distributions compared to the standard sampling model. This addresses an open question posed by Aliakbarpour et al. (2016).