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Discovering the latent structure from many observed variables is an importantyet challenging learning task. Existing approaches for discovering latentstructures often require the unknown number of hidden states as an input. Inthis paper, we propose a quartet based approach which is \emph{agnostic} tothis number. The key contribution is a novel rank characterization of thetensor associated with the marginal distribution of a quartet. Thischaracterization allows us to design a \emph{nuclear norm} based test forresolving quartet relations. We then use the quartet test as a subroutine in adivide-and-conquer algorithm for recovering the latent tree structure. Undermild conditions, the algorithm is consistent and its error probability decaysexponentially with increasing sample size. We demonstrate that the proposedapproach compares favorably to alternatives. In a real world stock dataset, italso discovers meaningful groupings of variables, and produces a model thatfits the data better.

Unfolding Latent Tree Structures using 4th Order Tensors