# Conditional Matrix Flows for Gaussian Graphical Models

arXiv: https://arxiv.org/abs/2306.07255 · HTML: https://mlsanity.com/paper/2306.07255/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 52 references that are in the map.
- First posted on arXiv 12 Jun 2023.

## Builds on (most cited first, 13 of 13)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Best Subset Selection via a Modern Optimization Lens](https://mlsanity.com/paper/1507.03133/) (2015)
- [Sylvester Normalizing Flows for Variational Inference](https://mlsanity.com/paper/1803.05649/) (2018)
- [Semi-Conditional Normalizing Flows for Semi-Supervised Learning](https://mlsanity.com/paper/1905.00505/) (2019)
- [The Graphical Horseshoe Estimator for Inverse Covariance Matrices](https://mlsanity.com/paper/1707.06661/) (2017)
- [Jacobian Determinant of Normalizing Flows](https://mlsanity.com/paper/2102.06539/) (2021)
- [Variational Full Bayes Lasso: Knots Selection in Regression Splines](https://mlsanity.com/paper/2102.13548/) (2021)
- [Bayesian Markov Blanket Estimation](https://mlsanity.com/paper/1510.01485/) (2015)

## Built on by (most cited first, 5 of 5)

- [Injective flows for star-like manifolds](https://mlsanity.com/paper/2406.09116/) (2024)
- [Probabilistic Iterative Hard Thresholding for Sparse Learning](https://mlsanity.com/paper/2409.01413/) (2024)
- [NIMO: a Nonlinear Interpretable MOdel](https://mlsanity.com/paper/2506.05059/) (2025)
- [Marginal Flow: a flexible and efficient framework for density estimation](https://mlsanity.com/paper/2509.26221/) (2025)
- [Dark Energy Survey Year 3 results: Simulation-based wCDM inference from weak lensing and galaxy clustering maps with deep learning: Analysis design](https://mlsanity.com/paper/2511.04681/) (2025)

## Abstract (arXiv)

Studying conditional independence among many variables with few observations is a challenging task. Gaussian Graphical Models (GGMs) tackle this problem by encouraging sparsity in the precision matrix through lq regularization with q≤1. However, most GMMs rely on the l1 norm because the objective is highly non-convex for sub-l1 pseudo-norms. In the frequentist formulation, the l1 norm relaxation provides the solution path as a function of the shrinkage parameter λ. In the Bayesian formulation, sparsity is instead encouraged through a Laplace prior, but posterior inference for different λ requires repeated runs of expensive Gibbs samplers. Here we propose a general framework for variational inference with matrix-variate Normalizing Flow in GGMs, which unifies the benefits of frequentist and Bayesian frameworks. As a key improvement on previous work, we train with one flow a continuum of sparse regression models jointly for all regularization parameters λ and all lq norms, including non-convex sub-l1 pseudo-norms. Within one model we thus have access to (i) the evolution of the posterior for any λ and any lq (pseudo-) norm, (ii) the marginal log-likelihood for model selection, and (iii) the frequentist solution paths through simulated annealing in the MAP limit.