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Aizenbud Y, Jaffe A, Wang M, Hu A, Amsel N, Nadler B, Chang JT, Kluger Y. Spectral top-down recovery of latent tree models. Inf inference 2023; 12:iaad032. [PMID: 37593361 PMCID: PMC10431953 DOI: 10.1093/imaiai/iaad032] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/10/2021] [Revised: 03/24/2023] [Accepted: 06/24/2023] [Indexed: 08/19/2023]
Abstract
Modeling the distribution of high-dimensional data by a latent tree graphical model is a prevalent approach in multiple scientific domains. A common task is to infer the underlying tree structure, given only observations of its terminal nodes. Many algorithms for tree recovery are computationally intensive, which limits their applicability to trees of moderate size. For large trees, a common approach, termed divide-and-conquer, is to recover the tree structure in two steps. First, separately recover the structure of multiple, possibly random subsets of the terminal nodes. Second, merge the resulting subtrees to form a full tree. Here, we develop spectral top-down recovery (STDR), a deterministic divide-and-conquer approach to infer large latent tree models. Unlike previous methods, STDR partitions the terminal nodes in a non random way, based on the Fiedler vector of a suitable Laplacian matrix related to the observed nodes. We prove that under certain conditions, this partitioning is consistent with the tree structure. This, in turn, leads to a significantly simpler merging procedure of the small subtrees. We prove that STDR is statistically consistent and bound the number of samples required to accurately recover the tree with high probability. Using simulated data from several common tree models in phylogenetics, we demonstrate that STDR has a significant advantage in terms of runtime, with improved or similar accuracy.
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Affiliation(s)
- Yariv Aizenbud
- Program in Applied Mathematics, Yale University, New Haven, CT 06511, USA
| | - Ariel Jaffe
- Program in Applied Mathematics, Yale University, New Haven, CT 06511, USA
| | - Meng Wang
- Department of Pathology, Yale University, New Haven, CT 06511, USA
| | - Amber Hu
- Program in Applied Mathematics, Yale University, New Haven, CT 06511, USA
| | - Noah Amsel
- Program in Applied Mathematics, Yale University, New Haven, CT 06511, USA
| | - Boaz Nadler
- Department of Computer Science, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Joseph T Chang
- Department of Statistics, Yale University, New Haven, CT 06520, USA
| | - Yuval Kluger
- Program in Applied Mathematics, Yale University, New Haven, CT 06511, USA
- Department of Pathology, Yale University, New Haven, CT 06511, USA
- Interdepartmental Program in Computational Biology and Bioinformatics, Yale University, New Haven, CT 06511, USA
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