We consider the problem of estimating high-dimensional Gaussian graphical
models corresponding to a single set of variables under several distinct
conditions. This problem is motivated by the task of recovering transcriptional
regulatory networks on the basis of gene expression data {containing
heterogeneous samples, such as different disease states, multiple species, or
different developmental stages}. We assume that most aspects of the conditional
dependence networks are shared, but that there are some structured differences
between them. Rather than assuming that similarities and differences between
networks are driven by individual edges, we take a node-based approach, which
in many cases provides a more intuitive interpretation of the network
differences. We consider estimation under two distinct assumptions: (1)
differences between the K networks are due to individual nodes that are
perturbed across conditions, or (2) similarities among the K networks are due
to the presence of common hub nodes that are shared across all K networks.
Using a row-column overlap norm penalty function, we formulate two convex
optimization problems that correspond to these two assumptions. We solve these
problems using an alternating direction method of multipliers algorithm, and we
derive a set of necessary and sufficient conditions that allows us to decompose
the problem into independent subproblems so that our algorithm can be scaled to
high-dimensional settings. Our proposal is illustrated on synthetic data, a
webpage data set, and a brain cancer gene expression data set.

该论文探究了高维高斯图形模型估计问题，提出了两种假设：网络差异源于被扰动的个别节点或在所有网络中共享的公共中心节点，然后使用凸优化问题和多重乘数算法解决，并以合成数据、网页数据和癌症基因表达数据为例进行说明。

基于节点学习的多个高斯图模型

Node-Based Learning of Multiple Gaussian Graphical Models

In this paper, we address the problem of embedded feature selection for
ranking on top of the list problems. We pose this problem as a regularized
empirical risk minimization with $p$-norm push loss function ($p=\infty$) and
sparsity inducing regularizers. We leverage the issues related to this
challenging optimization problem by considering an alternating direction method
of multipliers algorithm which is built upon proximal operators of the loss
function and the regularizer. Our main technical contribution is thus to
provide a numerical scheme for computing the infinite push loss function
proximal operator. Experimental results on toy, DNA microarray and BCI problems
show how our novel algorithm compares favorably to competitors for ranking on
top while using fewer variables in the scoring function.

本文通过使用 p 范数的推波损失函数以及稀疏正则化器，将嵌入式特征选择问题作为一种带约束的经验风险最小化问题进行研究，并提出了一种基于交替方向乘子法的数值方案，以计算无限推动损失函数的近端算子，通过在少数变量的情况下进行排名，我们与竞争对手相比得到了更好的实验结果。