# Ising Models with Latent Conditional Gaussian Variables

arXiv: https://arxiv.org/abs/1901.09712 · HTML: https://mlsanity.com/paper/1901.09712/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jan 2019.

## Built on by (most cited first, 2 of 2)

- [A Blueprint for Equilibrium-Based Differentiable Continuous-Variable Thermodynamic Computing](https://mlsanity.com/paper/2607.16183/) (2026)
- [Learning Gaussian Graphical Models with Latent Confounders](https://mlsanity.com/paper/2105.06600/) (2021)

## Abstract (arXiv)

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.