Certifiably Optimal Sparse Inverse Covariance Estimation
This work addresses the need for certifiably optimal sparse inverse covariance estimation in statistics and machine learning, offering a rigorous alternative to heuristic methods.
The paper tackles the problem of maximum likelihood estimation of sparse inverse covariance matrices by introducing a novel approach that solves the cardinality constrained likelihood problem to certifiable optimality, demonstrating it can handle dimensions up to 1,000s and produce significantly sparser solutions with fewer false discoveries while maintaining state-of-the-art accuracy.
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.