关于高斯图模型中估计的若干算法
On some algorithms for estimation in Gaussian graphical models
AI总结：
本文研究高斯图模型似然方程的迭代求解，比较避免大矩阵求逆的迭代比例缩放与零惩罚 graphical lasso 的邻域坐标下降算法，并为低色数图构造正定初值及简化 MLE 存在性证明。
摘要：
在高斯图模型中，似然方程通常必须通过迭代求解。我们研究了两种算法：一种是迭代比例缩放的版本，它避免了对大型矩阵求逆；另一种基于凸对偶，通过邻域坐标下降作用于协方差矩阵，对应于零惩罚的 graphical lasso。对于大型稀疏图，迭代比例缩放算法看起来是可行的，并具有简单的收敛性质。基于邻域坐标下降的算法速度极快，且对稀疏性的依赖较小，但需要一个正定初值才能收敛。我们给出了一种算法，用于为低色数图寻找这样的初值。因此，我们还得到了在此类情形下最大似然估计量存在性的简化证明。
英文摘要：
In Gaussian graphical models, the likelihood equations must typically be solved iteratively. We investigate two algorithms: A version of iterative proportional scaling which avoids inversion of large matrices, and an algorithm based on convex duality and operating on the covariance matrix by neighbourhood coordinate descent, corresponding to the graphical lasso with zero penalty. For large, sparse graphs, the iterative proportional scaling algorithm appears feasible and has simple convergence properties. The algorithm based on neighbourhood coordinate descent is extremely fast and less dependent on sparsity, but needs a positive definite starting value to converge. We give an algorithm for finding such a starting value for graphs with low colouring number. As a consequence, we also obtain a simplified proof for existence of the maximum likelihood estimator in such cases.
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