Fast Algorithms for Learning Latent Variables in Graphical Models
This work addresses a bottleneck in graphical model learning for researchers and practitioners, offering improved efficiency in latent variable estimation.
The paper tackles the problem of learning latent variables in Gaussian graphical models by focusing on estimating the low-rank component of the precision matrix, introducing fast, non-convex algorithms that match the best possible sample complexity and achieve computational speed-ups over existing methods.
We study the problem of learning latent variables in Gaussian graphical models. Existing methods for this problem assume that the precision matrix of the observed variables is the superposition of a sparse and a low-rank component. In this paper, we focus on the estimation of the low-rank component, which encodes the effect of marginalization over the latent variables. We introduce fast, proper learning algorithms for this problem. In contrast with existing approaches, our algorithms are manifestly non-convex. We support their efficacy via a rigorous theoretical analysis, and show that our algorithms match the best possible in terms of sample complexity, while achieving computational speed-ups over existing methods. We complement our theory with several numerical experiments.