The Exact Determinant of a Specific Class of Sparse Positive Definite Matrices

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.