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Exact Asymptotics for Learning Tree-Structured Graphical Models with Side Information: Noiseless and Noisy Samples

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Anshoo Tandon, Vincent Y. F. Tan, Shiyao Zhu

Machine Learning · Submitted 9 May 2020

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Given side information that an Ising tree-structured graphical model is homogeneous and has no external field, we derive the exact asymptotics of learning its structure from independently drawn samples.

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arXiv ID2005.04354 (version 1)
Pith ID4UBPHUCB, Pith's permanent record of this paper and its review
CitationAnshoo Tandon, Shiyao Zhu, Vincent Y. F. Tan. "Exact Asymptotics for Learning Tree-Structured Graphical Models with Side Information: Noiseless and Noisy Samples." arXiv:2005.04354 (2020). https://pith.science/paper/4UBPHUCB
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Given side information that an Ising tree-structured graphical model is homogeneous and has no external field, we derive the exact asymptotics of learning its structure from independently drawn samples. Our results, which leverage the use of probabilistic tools from the theory of strong large deviations, refine the large deviation (error exponents) results of Tan, Anandkumar, Tong, and Willsky [IEEE Trans. on Inform. Th., 57(3):1714--1735, 2011] and strictly improve those of Bresler and Karzand [Ann. Statist., 2020]. In addition, we extend our results to the scenario in which the samples are observed in random noise. In this case, we show that they strictly improve on the recent results of Nikolakakis, Kalogerias, and Sarwate [Proc. AISTATS, 1771--1782, 2019]. Our theoretical results demonstrate keen agreement with experimental results for sample sizes as small as that in the hundreds.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Given side information that an Ising tree-structured graphical model is homogeneous and has no external field, we derive the exact asymptotics of learning its structure from independently drawn samples. contribution 0.93 0.95 0.45 asserted 1.00
s1 Our results, which leverage the use of probabilistic tools from the theory of strong large deviations, refine the large deviation (error exponents) results of Tan, Anandkumar, Tong, and Willsky [IEEE Trans. on Inform. contribution 0.66 0.83 0.23 asserted 0.99
s2 Th., 57(3):1714--1735, 2011] and strictly improve those of Bresler and Karzand [Ann. Statist., 2020]. contribution 0.56 0.52 0.22 asserted 0.99
s3 In addition, we extend our results to the scenario in which the samples are observed in random noise. contribution 0.99 0.93 0.06 asserted 0.88
s4 In this case, we show that they strictly improve on the recent results of Nikolakakis, Kalogerias, and Sarwate [Proc. AISTATS, 1771--1782, 2019]. result 0.81 0.85 0.13 asserted 1.00
s5 Our theoretical results demonstrate keen agreement with experimental results for sample sizes as small as that in the hundreds. result 0.95 0.90 0.14 asserted 1.00

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

Given side information that an Ising tree-structured graphical model is homogeneous and has no external field, we derive the exact asymptotics of learning its structure from independently drawn samples. Our results, which leverage the use of probabilistic tools from the theory of strong large deviations, refine the large deviation (error exponents) results of Tan, Anandkumar, Tong, and Willsky [IEEE Trans. on Inform. Th., 57(3):1714--1735, 2011] and strictly improve those of Bresler and Karzand [Ann. Statist., 2020]. In addition, we extend our results to the scenario in which the samples are observed in random noise. In this case, we show that they strictly improve on the recent results of Nikolakakis, Kalogerias, and Sarwate [Proc. AISTATS, 1771--1782, 2019]. Our theoretical results demonstrate keen agreement with experimental results for sample sizes as small as that in the hundreds.

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  • Since its publication, researchers have continued to look into analyzing properties of this algorithm [ref] and generalizing it to other classes of graphs, e.g., polytrees [ref] , bounded treewidth graphs [ref] , and mixtures of trees [ref] .

    Near-Optimal Learning of Tree-Structured Distributions by Chow-Liu · Related Work · 0.80

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  • [ref] is to incorporate the multiplicative factor $$, which accounts for the number of $3$-node sub-trees of $_P$ that contribute to dominant errors [ref] in the ML learning algorithm.

    Optimal Rates for Learning Hidden Tree Structures · Strong Large Deviations: Exact Asymptotics · 0.85

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