Learning Markov graphs up to edit distance

This paper presents a rate distortion approach to Markov graph learning. It provides lower bounds on the number of samples required for any algorithm to learn the Markov graph structure of a probability distribution, up to edit distance. We first prove a general result for any probability distribution, and then specialize it for Ising and Gaussian models. In particular, for both Ising and Gaussian models on p variables with degree at most d, we show that at least Ω((d - s/p)log p) samples are required for any algorithm to learn the graph structure up to edit distance s. Our bounds represent a strong converse; i.e., we show that for a lower number of samples, the probability of error goes to 1 as the problem size increases. These results show that substantial gains in sample complexity may not be possible without paying a significant price in edit distance error.