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.

翻译：我们提出了一种新颖的拟牛顿方法，用于求解稀疏逆协方差估计问题（也称为图形最小绝对收缩与选择算子，GLASSO）。该问题通常通过二阶二次逼近求解，但在此类算法中，Hessian项结构复杂且计算成本高昂。因此，我们的方法将Hessian的逆矩阵作为预条件子，通过增强 \(\ell_1\) 元素的复杂性来简化和逼近二次项。预条件后的问题变量仅通过彼此的 \(\ell_1\) 次导数相互耦合，而该次导数可借助梯度本身以极低成本进行估计，从而使得算法可并行化并在GPU硬件加速器上高效实现。在合成数据与真实数据上的数值结果表明，我们的方法与其他最新技术相比具有竞争力。