# Predictive Learning on Hidden Tree-Structured Ising Models

arXiv: https://arxiv.org/abs/1812.04700 · HTML: https://mlsanity.com/paper/1812.04700/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Dec 2018.

## Built on by (most cited first, 7 of 7)

- [Chow-Liu++: Optimal Prediction-Centric Learning of Tree Ising Models](https://mlsanity.com/paper/2106.03969/) (2021)
- [SGA: A Robust Algorithm for Partial Recovery of Tree-Structured Graphical Models with Noisy Samples](https://mlsanity.com/paper/2101.08917/) (2021)
- [Robust estimation of tree structured models](https://mlsanity.com/paper/2102.05472/) (2021)
- [CoRMF: Criticality-Ordered Recurrent Mean Field Ising Solver](https://mlsanity.com/paper/2403.03391/) (2024)
- [Estimating Ising Models in Total Variation Distance](https://mlsanity.com/paper/2511.21008/) (2025)
- [Decentralized Learning of Tree-Structured Gaussian Graphical Models from Noisy Data](https://mlsanity.com/paper/2109.10642/) (2021)
- [Active-LATHE: An Active Learning Algorithm for Boosting the Error Exponent for Learning Homogeneous Ising Trees](https://mlsanity.com/paper/2110.14341/) (2021)

## Abstract (arXiv)

We provide high-probability sample complexity guarantees for exact structure recovery and accurate predictive learning using noise-corrupted samples from an acyclic (tree-shaped) graphical model. The hidden variables follow a tree-structured Ising model distribution, whereas the observable variables are generated by a binary symmetric channel taking the hidden variables as its input (flipping each bit independently with some constant probability q∈ [0,1/2)). In the absence of noise, predictive learning on Ising models was recently studied by Bresler and Karzand (2020); this paper quantifies how noise in the hidden model impacts the tasks of structure recovery and marginal distribution estimation by proving upper and lower bounds on the sample complexity. Our results generalize state-of-the-art bounds reported in prior work, and they exactly recover the noiseless case (q=0). In fact, for any tree with p vertices and probability of incorrect recovery δ>0, the sufficient number of samples remains logarithmic as in the noiseless case, i.e., O(log(p/δ)), while the dependence on q is Obig( 1/(1-2q)4 big), for both aforementioned tasks. We also present a new equivalent of Isserlis' Theorem for sign-valued tree-structured distributions, yielding a new low-complexity algorithm for higher-order moment estimation.