We provide high probability finite sample complexity guarantees for hidden non-parametric structure learning of tree-shaped graphical models, whose hidden and observable nodes are discrete random variables with either finite or countable alphabets. We study a fundamental quantity called the (noisy) information threshold, which arises naturally from the error analysis of the Chow-Liu algorithm and, as we discuss, provides explicit necessary and sufficient conditions on sample complexity, by effectively summarizing the difficulty of the tree-structure learning problem. Specifically, we show that the finite sample complexity of the Chow-Liu algorithm for ensuring exact structure recovery from noisy data is inversely proportional to the information threshold squared (provided it is positive), and scales almost logarithmically relative to the number of nodes over a given probability of failure. Conversely, we show that, if the number of samples is less than an absolute constant times the inverse of information threshold squared, then no algorithm can recover the hidden tree structure with probability greater than one half. As a consequence, our upper and lower bounds match with respect to the information threshold, indicating that it is a fundamental quantity for the problem of learning hidden tree-structured models. Further, the Chow-Liu algorithm with noisy data as input achieves the optimal rate with respect to the information threshold. Lastly, as a byproduct of our analysis, we resolve the problem of tree structure learning in the presence of non-identically distributed observation noise, providing conditions for convergence of the Chow-Liu algorithm under this setting, as well.
Spectral Methods for Learning Multivariate Latent Tree Structure
UC Irvine（加州大学欧文分校）；UC San Diego（加州大学圣迭戈分校）；Microsoft Research New England（微软新英格兰研究院）；Wharton School, University of Pennsylvania（宾夕法尼亚大学沃顿商学院）；Carnegie Mellon University（卡内基梅隆大学）；Rutgers University（罗格斯大学）
An efficient solution to Hidden Markov Models on trees with coupled branches
Center of Mathematical Sciences and Applications, Harvard University（哈佛大学数学科学与应用中心）；Dana-Farber Cancer Institute（丹娜法伯癌症研究院）；Harvard Medical School（哈佛医学院）；Broad Institute of MIT and Harvard（麻省理工学院与哈佛大学布罗德研究所）
MurTree: Optimal Classification Trees via Dynamic Programming and Search
Delft University of Technology（代尔夫特理工大学）；LAAS CNRS（法国国家科学研究中心LAAS实验室）；RMIT University（皇家墨尔本理工大学）；University of Melbourne（墨尔本大学）；Monash University（莫纳什大学）；DATA61（DATA61研究所）
