Se Young Kim

Jun Ho Yoon, Seyoung Kim

In many real-world problems, complex dependencies are present both among samples and among features. The Kronecker sum or the Cartesian product of two graphs, each modeling dependencies across features and across samples, has been used as an inverse covariance matrix for a matrix-variate Gaussian distribution, as an alternative to a Kronecker-product inverse covariance matrix, due to its more intuitive sparse structure. However, the existing methods for sparse Kronecker-sum inverse covariance estimation are limited in that they do not scale to more than a few hundred features and samples and that the unidentifiable parameters pose challenges in estimation. In this paper, we introduce EiGLasso, a highly scalable method for sparse Kronecker-sum inverse covariance estimation, based on Newton's method combined with eigendecomposition of the two graphs for exploiting the structure of Kronecker sum. EiGLasso further reduces computation time by approximating the Hessian based on the eigendecomposition of the sample and feature graphs. EiGLasso achieves quadratic convergence with the exact Hessian and linear convergence with the approximate Hessian. We describe a simple new approach to estimating the unidentifiable parameters that generalizes the existing methods. On simulated and real-world data, we demonstrate that EiGLasso achieves two to three orders-of-magnitude speed-up compared to the existing methods.

Calvin McCarter, Seyoung Kim

This paper addresses the problem of scalable optimization for L1-regularized conditional Gaussian graphical models. Conditional Gaussian graphical models generalize the well-known Gaussian graphical models to conditional distributions to model the output network influenced by conditioning input variables. While highly scalable optimization methods exist for sparse Gaussian graphical model estimation, state-of-the-art methods for conditional Gaussian graphical models are not efficient enough and more importantly, fail due to memory constraints for very large problems. In this paper, we propose a new optimization procedure based on a Newton method that efficiently iterates over two sub-problems, leading to drastic improvement in computation time compared to the previous methods. We then extend our method to scale to large problems under memory constraints, using block coordinate descent to limit memory usage while achieving fast convergence. Using synthetic and genomic data, we show that our methods can solve one million dimensional problems to high accuracy in a little over a day on a single machine.

Xi Chen, Qihang Lin, Seyoung Kim et al.

We study the problem of learning high dimensional regression models regularized by a structured-sparsity-inducing penalty that encodes prior structural information on either input or output sides. We consider two widely adopted types of such penalties as our motivating examples: 1) overlapping group lasso penalty, based on the l1/l2 mixed-norm penalty, and 2) graph-guided fusion penalty. For both types of penalties, due to their non-separability, developing an efficient optimization method has remained a challenging problem. In this paper, we propose a general optimization approach, called smoothing proximal gradient method, which can solve the structured sparse regression problems with a smooth convex loss and a wide spectrum of structured-sparsity-inducing penalties. Our approach is based on a general smoothing technique of Nesterov. It achieves a convergence rate faster than the standard first-order method, subgradient method, and is much more scalable than the most widely used interior-point method. Numerical results are reported to demonstrate the efficiency and scalability of the proposed method.