[Paper Review] A Pathwise Algorithm for Covariance Selection

V. Krishnamurthy, Alexandre d’Aspremont|arXiv (Cornell University)|Aug 2, 2009

Face and Expression Recognition4 citations

TL;DR

This paper proposes a pathwise algorithm for covariance selection using numerical continuation to efficiently compute the full regularization path of sparse inverse covariance matrices. By combining predictor-corrector steps with a coordinate descent method that solves cubic equations in closed form, the method reduces computational cost significantly compared to solving each regularization parameter independently.

ABSTRACT

Covariance selection seeks to estimate a covariance matrix by maximum likelihood while restricting the number of nonzero inverse covariance matrix coefficients. A single penalty parameter usually controls the tradeoff between log likelihood and sparsity in the inverse matrix. We describe an efficient algorithm for computing a full regularization path of solutions to this problem.

Motivation & Objective

• To address the computational challenge of computing the full regularization path for sparse inverse covariance estimation.
• To reduce the cost of solving multiple instances of the covariance selection problem for different penalty parameters.
• To develop a scalable and efficient method that leverages warm-starting and path-following techniques for large-scale problems.
• To enable practical application of covariance selection in high-dimensional settings where cross-validation over multiple penalty values is required.
• To extend the framework to online settings where the covariance matrix is updated incrementally.

Proposed method

• Formulates the covariance selection problem as a convex optimization problem using an ℓ1 penalty on the inverse covariance matrix.
• Applies numerical continuation via a predictor-corrector method to trace the regularization path from high to low penalty values.
• Uses a predictor step based on solving a large structured linear system via conjugate gradient to estimate the next solution along the path.
• Employs a corrector step using a coordinate descent algorithm where each update involves solving a cubic equation in closed form.
• Derives a dual formulation and uses barrier methods to define a central path that guides the path-following procedure.
• Adapts the algorithm for online covariance selection by updating the solution efficiently when the sample covariance matrix changes incrementally.

Experimental results

Research questions

• RQ1 How can the full regularization path of sparse inverse covariance matrices be computed efficiently without solving separate optimization problems for each penalty parameter?
• RQ2 What is the computational advantage of using predictor-corrector continuation over solving individual instances of the covariance selection problem?
• RQ3 Can a coordinate descent method with closed-form updates be effectively used in the corrector step of a pathwise algorithm?
• RQ4 How does the algorithm perform in terms of convergence speed and sparsity recovery on synthetic and real-world data?
• RQ5 To what extent can the algorithm be adapted for online learning scenarios with incremental updates to the covariance matrix?

Key findings

• The pathwise algorithm significantly reduces computational cost by reusing information from previous solutions via warm-starting and path-following.
• The corrector step uses a coordinate descent method where each update is computed in closed form by solving a cubic equation, improving efficiency.
• The predictor step relies on solving a large linear system via conjugate gradient, which is computationally efficient when well-conditioned.
• Numerical experiments show that the algorithm scales well with dimension and produces accurate sparse inverse covariance estimates.
• The online variant of the algorithm, using a single predictor-corrector step with k=1, achieves fast convergence after covariance updates, making it suitable for large-scale or streaming data.