发表机构
IMAG, Université de Montpellier; LEMON, Inria（蒙彼利埃大学 IMAG; Inria LEMON）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对多元极值中 Hüsler-Reiss 精度矩阵的块结构，提出基于凸融合惩罚的正则化框架，可同时估计系数与变量划分，并证明一致性，实验验证其有效性。
AI 中文摘要
估计 Hüsler-Reiss 精度矩阵是多元极值统计推断中的一个基本问题。在高维设置下，未知参数的数量随维度呈二次增长，这使得正则化不可或缺。现有方法通过利用稀疏性来正则化估计问题。在本文中，我们考虑一种替代的结构假设，即 Hüsler-Reiss 精度矩阵具有块结构。为了估计这样的矩阵，我们引入了一种基于凸融合惩罚的新正则化框架。通过鼓励行和列合并，该方法提供了精度矩阵的简约表示，允许同时估计其系数和变量的潜在划分。由此产生的凸优化问题通过一种结合基于梯度的更新与渐进融合步骤的高效算法求解。我们为进入惩罚项的经验权重建立了非渐近集中界，并在适当的正则性条件下证明了块恢复和精度矩阵估计的一致性。数值实验表明，我们的方法在各种配置下能够准确恢复潜在块结构，同时准确估计 Hüsler-Reiss 精度矩阵，展示了基于融合的正则化对多元极值的实际益处。通过将所提方法应用于外汇数据，这些益处也得到了展示。
英文摘要
Estimating the H{ü}sler-Reiss precision matrix is a fundamental problem for statistical inference in multivariate extremes. In high-dimensional settings, the number of unknown parameters grows quadratically with the dimension, making regularisation indispensable. Existing approaches regularise the estimation problem by exploiting sparsity. In this paper, we consider an alternative structural assumption, namely that the H{ü}sler-Reiss precision matrix is block-structured. To estimate such a matrix, we introduce a new regularisation framework based on a convex fusion penalty. By encouraging rows and columns to merge, this approach provides a parsimonious representation of the precision matrix, allowing for the simultaneous estimation of its coefficients and the underlying partition of the variables. The resulting convex optimisation problem is solved by an efficient algorithm combining gradient-based updates with progressive fusion steps. We establish non-asymptotic concentration bounds for the empirical weights entering the penalty and prove consistency of both block recovery and precision matrix estimation under suitable regularity conditions. Numerical experiments demonstrate that our methodology accurately recovers the latent block structure while accurately estimating the H{ü}sler-Reiss precision matrix across various configurations, illustrating the practical benefits of fusion-based regularisation for multivariate extremes. These benefits are also demonstrated by applying the proposed method to foreign exchange data.
