Learning Latent Tree Graphical Models (1009.2722v1)

Abstract: 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.

• The paper introduces two novel algorithms—Recursive Grouping and CLGrouping—for constructing minimal latent tree models from partially observed data.
• It establishes structural consistency and efficiency guarantees, achieving risk consistency for Gaussian and discrete symmetric distributions.
• Empirical evaluations on synthetic and real-world datasets demonstrate superior performance over traditional methods in finance and text analysis.

Overview of "Learning Latent Tree Graphical Models"

The paper by Choi, Tan, Anandkumar, and Willsky addresses the problem of learning latent tree graphical models from data sets where observations are available for only a subset of variables. Unlike traditional approaches, the paper focuses on methods that do not restrict observed variables to be leaf nodes and seeks to construct minimal latent trees, which eliminate redundant hidden nodes.

Key Contributions

1. Algorithm Development: The paper introduces two main algorithms: Recursive Grouping and CLGrouping.
   • Recursive Grouping (RG): This algorithm incrementally builds the latent tree structure from the observed nodes by identifying sibling groups through calculated information distances, iteratively constructing the tree from the bottom up.
   • CLGrouping: This involves a two-step process, initially constructing a Chow-Liu tree over the observed variables, followed by refined grouping procedures that focus on smaller subsets. CLGrouping further enhances efficiency and accuracy in tree learning compared to RG, especially for trees with large diameters.
2. Consistency and Complexity: Both algorithms are shown to be structurally consistent; they are equipped with performance guarantees in terms of both computational and sample complexity. For Gaussian and certain discrete symmetric distributions, risk consistency is also achieved, ensuring convergence of parameter estimates.
3. Empirical Evaluation: The authors provide extensive numerical experiments using synthetic data sets, such as hidden Markov models, to demonstrate that their proposed methods generally outperform existing approaches like neighbor-joining (NJ) under various configurations.
4. Real-world Applications: The paper extends its methodology to real-world data applications, modeling the dependency structures of stock returns and newsgroup word occurrences. These applications showcase the practical utility of the proposed models in capturing and simplifying complex dependency structures.

Theoretical and Practical Implications

Theoretical Advancements:

This research advances the theoretical understanding of latent tree models by resolving key challenges in structure learning, particularly the positioning and identification of hidden nodes without redundant nodes.

Practical Applications:

Beyond theoretical contributions, practical implications include enhanced model accuracy and reduced computational overhead in real-world applications such as finance and text processing, where understanding dependencies between variables is critical.

Future Directions

The proposed work leaves room for several future research avenues:

• Scalability and Automation: While the algorithms have been demonstrated to work effectively with a significant number of variables, extending their scalability to large-scale data sets remains promising.
• Robustness in Real-world Noisy Data: Exploring robust modifications that further mitigate the effect of noisy or sparse observations could enhance their usability.
• Extension to Non-tree Models: While focusing on tree structures, there is potential to investigate transformations or expansions to more general network topologies.

Overall, the paper presents a comprehensive approach to learning latent tree models, balancing theoretical insight with empirical validation and practical applicability.