Motivated by the important statistical role of sparsity, the paper uncovers four reparametrizations for covariance matrices in which sparsity is associated with conditional independence graphs in a notional Gaussian model. The intimate relationship between the Iwasawa decomposition of the general linear group and the open cone of positive definite matrices allows a unifying perspective. Specifically, the positive definite cone can be reconstructed without loss or redundancy from the exponential map applied to four Lie subalgebras determined by the Iwasawa decomposition of the general linear group. This accords geometric interpretations to the reparametrizations and the corresponding notion of sparsity. Conditions that ensure legitimacy of the reparametrizations for statistical models are identified. While the focus of this work is on understanding population-level structure, there are strong methodological implications. In particular, since the population-level sparsity manifests in a vector space, imposition of sparsity on relevant sample quantities produces a covariance estimate that respects the positive definite cone constraint.

翻译：受稀疏性在统计学中重要作用的启发，本文揭示了协方差矩阵的四种重参数化方法，其中稀疏性与一个虚拟高斯模型中的条件独立图相关联。一般线性群的Iwasawa分解与正定矩阵开锥之间的密切关系提供了统一视角。具体而言，正定锥可以通过对由一般线性群的Iwasawa分解确定的四个李子代数应用指数映射而被无损失或冗余地重构。这为上述重参数化及相应的稀疏性概念赋予了几何解释。本文识别了确保这些重参数化在统计模型中合法性的条件。本研究的重点在于理解群体层面的结构，但具有重要的方法论启示。特别是，由于群体层面的稀疏性在向量空间中体现，对相关样本量施加稀疏性可产生一个尊重正定锥约束的协方差估计。