带 Golazo 惩罚的潜在高斯与 Hüsler–Reiss 图模型
Latent Gaussian and Hüsler--Reiss Graphical Models with Golazo Penalty
AI总结：
本文提出用灵活的 Golazo 惩罚统一推广潜在高斯与 Hüsler–Reiss 图模型的结构学习，可涵盖自适应 lasso、正依赖约束和预定稀疏模式，并开发算法在模拟与真实数据上验证。
摘要：
实际问题中潜在变量的存在很常见，例如某些变量难以或昂贵测量，或者干脆未知。当潜在变量未被考虑时，高斯图模型的结构学习会因潜在变量引入的观测变量间额外相关而变得模糊。针对该问题的标准方法是图 lasso 的潜在版本，它将逆协方差矩阵分解为稀疏部分和低秩部分，并分别进行惩罚。该方法最近被成功扩展到 Hüsler–Reiss 图模型，后者可视为极值统计中高斯图模型的类比。本文提出通过灵活的 Golazo 惩罚对高斯和 Hüsler–Reiss 图模型的结构学习进行推广。这使我们能够引入例如自适应 lasso、正依赖约束或预定稀疏模式及其组合的潜在版本。我们为带 Golazo 惩罚的两种潜在图模型开发了算法，并在模拟数据和真实数据上进行了演示。
英文摘要：
The existence of latent variables in practical problems is common, for example when some variables are difficult or expensive to measure, or simply unknown. When latent variables are unaccounted for, structure learning for Gaussian graphical models can be blurred by additional correlation between the observed variables that is incurred by the latent variables. A standard approach for this problem is a latent version of the graphical lasso that splits the inverse covariance matrix into a sparse and a low-rank part that are penalized separately. This approach has recently been extended successfully to Hüsler--Reiss graphical models, which can be considered as an analogue of Gaussian graphical models in extreme value statistics. In this paper we propose a generalization of structure learning for Gaussian and Hüsler--Reiss graphical models via the flexible Golazo penalty. This allows us to introduce latent versions of for example the adaptive lasso, positive dependence constraints or predetermined sparsity patterns, and combinations of those. We develop algorithms for both latent graphical models with the Golazo penalty and demonstrate them on simulated and real data.
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