Optimal estimation of Gaussian (poly)trees
This work provides optimal methods for learning Gaussian graphical models, which is important for statistical inference and machine learning applications, though it is incremental as it builds on existing algorithms like Chow-Liu and PC.
The paper tackles the problem of learning Gaussian trees and polytrees from data, developing optimal algorithms for distribution and structure learning with explicit finite-sample guarantees and matching lower bounds.
We develop optimal algorithms for learning undirected Gaussian trees and directed Gaussian polytrees from data. We consider both problems of distribution learning (i.e. in KL distance) and structure learning (i.e. exact recovery). The first approach is based on the Chow-Liu algorithm, and learns an optimal tree-structured distribution efficiently. The second approach is a modification of the PC algorithm for polytrees that uses partial correlation as a conditional independence tester for constraint-based structure learning. We derive explicit finite-sample guarantees for both approaches, and show that both approaches are optimal by deriving matching lower bounds. Additionally, we conduct numerical experiments to compare the performance of various algorithms, providing further insights and empirical evidence.