Spectral Methods for Learning Multivariate Latent Tree Structure

This work considers the problem of learning the structure of multivariate linear tree models, whichinclude a variety of directed tree graphical models with continuous, discrete, and mixed latent variablessuch as linear-Gaussian models, hidden Markov models, Gaussian mixture models, and Markov evolu-tionary trees. The setting is one where we only have samples from certain observed variables in the tree,and our goal is to estimate the tree structure (i.e., the graph of how the underlying hidden variables areconnected to each other and to the observed variables). We propose the Spectral Recursive Grouping al-gorithm, an efficient and simple bottom-up procedure for recovering the tree structure from independentsamples of the observed variables. Our finite sample size bounds for exact recovery of the tree structurereveal certain natural dependencies on underlying statistical and structural properties of the underlyingjoint distribution. Furthermore, our sample complexity guarantees have no explicit dependence on thedimensionality of the observed variables, making the algorithm applicable to many high-dimensional set-tings. At the heart of our algorithm is a spectral quartet test for determining the relative topology of aquartet of variables from second-order statistics.