pISTA: preconditioned Iterative Soft Thresholding Algorithm for Graphical Lasso

TL;DR

The paper introduces pISTA, a preconditioned iterative soft thresholding algorithm for graphical lasso that improves computational efficiency and parallelization, especially on GPU hardware, by approximating the Hessian inverse.

Contribution

It proposes a novel quasi-Newton method using Hessian inverse as a preconditioner for sparse inverse covariance estimation, enabling efficient parallel computation.

Findings

Competitive performance on synthetic data

Effective GPU acceleration

Simplified quadratic approximation

Abstract

We propose a novel quasi-Newton method for solving the sparse inverse covariance estimation problem also known as the graphical least absolute shrinkage and selection operator (GLASSO). This problem is often solved using a second-order quadratic approximation. However, in such algorithms the Hessian term is complex and computationally expensive to handle. Therefore, our method uses the inverse of the Hessian as a preconditioner to simplify and approximate the quadratic element at the cost of a more complex \(\ell_1\) element. The variables of the resulting preconditioned problem are coupled only by the \(\ell_1\) sub-derivative of each other, which can be guessed with minimal cost using the gradient itself, allowing the algorithm to be parallelized and implemented efficiently on GPU hardware accelerators. Numerical results on synthetic and real data demonstrate that our method is competitive with other state-of-the-art approaches.