Hierarchical Tensor Decomposition of Latent Tree Graphical Models.

We approach the problem of estimating the parameters of a latent tree graphical model from a hierarchical tensor decomposition point of view. In this new view, the marginal probability table of the observed variables is treated as a tensor, and we show that: (i) the latent variables induce low rank structures in various matricizations of the tensor; (ii) this collection of low rank matricizations induces a hierarchical low rank decomposition of the tensor. We further derive an optimization problem for estimating (alternative) parameters of a latent tree graphical model, allowing us to represent the marginal probability table of the observed variables in a compact and robust way. The optimization problem aims to find the best hierarchical low rank approximation of a tensor in Frobenius norm. For correctly specified latent tree graphical models, we show that a global optimum of the optimization problem can be obtained via a recursive decomposition algorithm. This algorithm recovers previous spectral algorithms for hidden Markov models (Hsu et al., 2009; Foster et al., 2012) and latent tree graphical models (Parikh et al., 2011; Song et al., 2011) as special cases, elucidating the global objective these algorithms are optimizing. For misspecified latent tree graphical models, we derive a novel decomposition based on our framework, and provide approximation guarantee and computational complexity analysis. In both synthetic and real world data, this new estimator significantly improves over the state-of-the-art.