For a specific class of sparse Gaussian graphical models, we provide a closed-form solution for the determinant of the covariance matrix. In our framework, the graphical interaction model (i.e., the covariance selection model) is equal to replacement product of $\mathcal{K}_{n}$ and $\mathcal{K}_{n-1}$, where $\mathcal{K}_n$ is the complete graph with $n$ vertices. Our analysis is based on taking the Fourier transform of the local factors of the model, which can be viewed as an application of the Normal Factor Graph Duality Theorem and holographic algorithms. The closed-form expression is obtained by applying the Matrix Determinant Lemma on the transformed graphical model. In this context, we will also define a notion of equivalence between two Gaussian graphical models.

翻译：针对一类稀疏高斯图模型，我们给出了协方差矩阵行列式的闭式解。在该框架下，图形交互模型（即协方差选择模型）等价于$\mathcal{K}_{n}$与$\mathcal{K}_{n-1}$的替代积，其中$\mathcal{K}_n$为具有$n$个顶点的完全图。我们的分析基于对模型局部因子进行傅里叶变换，这可视为正态因子图对偶定理与全息算法的应用。通过将矩阵行列式引理应用于变换后的图模型，得到了闭式表达式。在此背景下，我们还将定义两个高斯图模型之间的等价概念。