[Paper Review] Ising Models with Latent Conditional Gaussian Variables

Frank Nussbaum, Joachim Giesen|arXiv (Cornell University)|Jan 28, 2019

Bayesian Modeling and Causal Inference22 references4 citations

TL;DR

This paper proposes a convex optimization framework for learning Ising models with latent conditional Gaussian variables by decomposing the interaction parameters into sparse and low-rank components. The method uses a regularized likelihood approach that jointly promotes sparsity and low-rank structure, enabling consistent estimation in high-dimensional settings where latent variables induce indirect interactions.

ABSTRACT

Ising models describe the joint probability distribution of a vector of binary feature variables. Typically, not all the variables interact with each other and one is interested in learning the presumably sparse network structure of the interacting variables. However, in the presence of latent variables, the conventional method of learning a sparse model might fail. This is because the latent variables induce indirect interactions of the observed variables. In the case of only a few latent conditional Gaussian variables these spurious interactions contribute an additional low-rank component to the interaction parameters of the observed Ising model. Therefore, we propose to learn a sparse + low-rank decomposition of the parameters of an Ising model using a convex regularized likelihood problem. We show that the same problem can be obtained as the dual of a maximum-entropy problem with a new type of relaxation, where the sample means collectively need to match the expected values only up to a given tolerance. The solution to the convex optimization problem has consistency properties in the high-dimensional setting, where the number of observed binary variables and the number of latent conditional Gaussian variables are allowed to grow with the number of training samples.

Motivation & Objective

• To address the failure of standard sparse Ising model learning when latent conditional Gaussian variables induce spurious indirect interactions among observed binary variables.
• To model the induced interactions as a low-rank component in the interaction matrix of the observed Ising model.
• To develop a convex regularized likelihood optimization problem that simultaneously promotes sparsity and low-rank structure in the interaction parameters.
• To establish consistency of the estimator in high-dimensional regimes where the number of observed variables and latent variables grow with sample size.
• To provide a dual interpretation via a maximum-entropy principle with a spectral norm relaxation, linking the method to information-theoretic principles.

Proposed method

• Formulates the problem as a convex regularized log-likelihood maximization with dual variables S (sparse) and L1−L2 (low-rank), subject to positive semidefiniteness constraints.
• Uses a spectral norm relaxation of the moment-matching constraint, replacing individual tolerance bounds with collective spectral tolerance λ.
• Derives the dual problem as maximizing ℓ(S + L1 − L2) − c‖S‖₁ − λ·tr(L1 + L2) with L1, L2 ⪰ 0, where ℓ is the log-likelihood function.
• Interprets the solution as a decomposition of the interaction matrix into a sparse component (direct interactions) and a low-rank component (indirect interactions via latent variables).
• Establishes equivalence to a maximum-entropy problem with a relaxed spectral norm constraint, providing a principled statistical foundation.
• Employs a marginal conditional Gaussian model to derive the relationship between latent variables and observed Ising parameters, showing that the induced interaction matrix has low-rank structure.

Experimental results

Research questions

• RQ1 Can a convex optimization framework effectively separate direct interactions from indirect interactions caused by a small number of latent conditionally Gaussian variables in an Ising model?
• RQ2 Does a sparse + low-rank decomposition of the interaction matrix lead to consistent estimation in high-dimensional settings?
• RQ3 How does the spectral norm relaxation of the moment-matching constraint relate to the dual regularized log-likelihood problem?
• RQ4 What is the statistical interpretation of the low-rank component in terms of latent variable-induced dependencies?
• RQ5 Can the proposed method be derived as a dual of a maximum-entropy problem with a novel relaxation type?

Key findings

• The method successfully models indirect interactions from latent conditionally Gaussian variables as a low-rank component in the interaction matrix of the observed Ising model.
• The proposed convex regularized likelihood problem achieves consistency in high-dimensional settings where both the number of observed binary variables and latent variables grow with the sample size.
• The dual problem reveals that the solution corresponds to a sparse + low-rank decomposition of the interaction parameters, with the low-rank part arising from the spectral norm relaxation.
• The spectral norm relaxation leads to a dual formulation with a trace penalty on a positive semidefinite matrix, promoting low-rank structure in the solution.
• The method is grounded in a maximum-entropy principle with a novel relaxation, linking it to information-theoretic principles and providing a principled alternative to strict moment matching.
• The marginal conditional Gaussian model derivation confirms that the induced interaction matrix has a low-rank structure proportional to RᵀΛ⁻¹R, justifying the low-rank assumption.