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### Learning Latent Tree Graphical Models

We study the problem of learning a latent tree graphical model where samples are available only from a subset of variables. We propose two consistent and computationally efficient algorithms for learning minimal latent trees, that is, trees without any redundant hidden nodes. Unlike many existing methods, the observed nodes (or variables) are not constrained to be leaf nodes. Our first algorithm, recursive grouping, builds the latent tree recursively by identifying sibling groups using so-called information distances. One of the main contributions of this work is our second algorithm, which we refer to as CLGrouping. CLGrouping starts with a pre-processing procedure in which a tree over the observed variables is constructed. This global step groups the observed nodes that are likely to be close to each other in the true latent tree, thereby guiding subsequent recursive grouping (or equivalent procedures) on much smaller subsets of variables. This results in more accurate and efficient learning of latent trees. We also present regularized versions of our algorithms that learn latent tree approximations of arbitrary distributions. We compare the proposed algorithms to other methods by performing extensive numerical experiments on various latent tree graphical models such as hidden Markov models and star graphs. In addition, we demonstrate the applicability of our methods on real-world datasets by modeling the dependency structure of monthly stock returns in the S&P index and of the words in the 20 newsgroups dataset.

### Conditional Chow-Liu Tree Structures for Modeling Discrete-Valued Vector Time Series

We consider the problem of modeling discrete-valued vector time series data using extensions of Chow-Liu tree models to capture both dependencies across time and dependencies across variables. Conditional Chow-Liu tree models are introduced, as an extension to standard Chow-Liu trees, for modeling conditional rather than joint densities. We describe learning algorithms for such models and show how they can be used to learn parsimonious representations for the output distributions in hidden Markov models. These models are applied to the important problem of simulating and forecasting daily precipitation occurrence for networks of rain stations. To demonstrate the effectiveness of the models, we compare their performance versus a number of alternatives using historical precipitation data from Southwestern Australia and the Western United States. We illustrate how the structure and parameters of the models can be used to provide an improved meteorological interpretation of such data.

### Forest Density Estimation

We study graph estimation and density estimation in high dimensions, using a family of density estimators based on forest structured undirected graphical models. For density estimation, we do not assume the true distribution corresponds to a forest; rather, we form kernel density estimates of the bivariate and univariate marginals, and apply Kruskal's algorithm to estimate the optimal forest on held out data. We prove an oracle inequality on the excess risk of the resulting estimator relative to the risk of the best forest. For graph estimation, we consider the problem of estimating forests with restricted tree sizes. We prove that finding a maximum weight spanning forest with restricted tree size is NP-hard, and develop an approximation algorithm for this problem. Viewing the tree size as a complexity parameter, we then select a forest using data splitting, and prove bounds on excess risk and structure selection consistency of the procedure. Experiments with simulated data and microarray data indicate that the methods are a practical alternative to Gaussian graphical models.

### Tree-dependent Component Analysis

We present a generalization of independent component analysis (ICA), where instead of looking for a linear transform that makes the data components independent, we look for a transform that makes the data components well fit by a tree-structured graphical model. Treating the problem as a semiparametric statistical problem, we show that the optimal transform is found by minimizing a contrast function based on mutual information, a function that directly extends the contrast function used for classical ICA. We provide two approximations of this contrast function, one using kernel density estimation, and another using kernel generalized variance. This tree-dependent component analysis framework leads naturally to an efficient general multivariate density estimation technique where only bivariate density estimation needs to be performed.

### Non-Parametric Structure Learning on Hidden Tree-Shaped Distributions

We provide high probability sample complexity guarantees for non-parametric structure learning of tree-shaped graphical models whose nodes are discrete random variables with a finite or countable alphabet, both in the noiseless and noisy regimes. First, we introduce a new, fundamental quantity called the (noisy) information threshold, which arises naturally from the error analysis of the Chow-Liu algorithm and characterizes not only the sample complexity, but also the inherent impact of the noise on the structure learning task, without explicit assumptions on the distribution of the model. This allows us to present the first non-parametric, high-probability finite sample complexity bounds on tree-structure learning from potentially noise-corrupted data. In particular, for number of nodes $p$, success rate $1-\delta$, and a fixed value of the information threshold, our sample complexity bounds for exact structure recovery are of the order of $\mathcal{O}\big(\log^{1+\zeta} (p/\delta)\big)$, for all $\zeta>0$, for both noiseless and noisy settings. Subsequently, we apply our results on two classes of hidden models, namely, the $M$-ary erasure channel and the generalized symmetric channel, illustrating the usefulness and importance of our framework. As a byproduct of our analysis, this paper resolves the open problem of tree structure learning in the presence of non-identically distributed observation noise, providing explicit conditions on the convergence of the Chow-Liu algorithm under this setting as well.