# Certifiably Optimal Sparse Inverse Covariance Estimation

## TL;DR

This paper introduces a novel method for sparse inverse covariance estimation that guarantees optimality and produces sparser, more accurate solutions compared to existing heuristics, even in high-dimensional settings.

## Contribution

It presents a new approach combining mixed-integer and convex optimization to solve the cardinality constrained likelihood problem with certifiable optimality.

## Key findings

- Successfully solves problems with inverse covariance matrices up to thousands of dimensions.
- Produces significantly sparser solutions than Glasso and other methods.
- Maintains state-of-the-art accuracy with fewer false discoveries.

## Abstract

We consider the maximum likelihood estimation of sparse inverse covariance matrices. We demonstrate that current heuristic approaches primarily encourage robustness, instead of the desired sparsity. We give a novel approach that solves the cardinality constrained likelihood problem to certifiable optimality. The approach uses techniques from mixed-integer optimization and convex optimization, and provides a high-quality solution with a guarantee on its suboptimality, even if the algorithm is terminated early. Using a variety of synthetic and real datasets, we demonstrate that our approach can solve problems where the dimension of the inverse covariance matrix is up to 1,000s. We also demonstrate that our approach produces significantly sparser solutions than Glasso and other popular learning procedures, makes less false discoveries, while still maintaining state-of-the-art accuracy.