一种高阶费歇耳信息矩阵分布式求逆方法

To obtain the explicit expression of Cramér-Rao lower bound (CRLB) and simplify the inversion process of higher-order Fisher information matrices (FIM) for undirected network navigation of multi-agent systems (MAS) with arbitrary information coupling, we proposed a distributed inversion method named block diagonalization method. Firstly, a high-order FIM was constructed for the arbitrary coupling undirected network navigation and positioning model, and then on the basis of its symmetry characteristics and the relationship between submatrices and network topology, the FIM was expressed as a linear combination of two block diagonal matrices, which contained the Fisher information of all nodes and edges respectively. Secondly, the matrix inversion lemma was used twice to derive the explicit expression of the inverse of FIM, i. e. CRLB and the equivalent FIM for network nodes. An iterative process was also introduced to decompose the whole inversion process into computations between several lower-order matrices, so as to mitigate the computational burden at every step. Finally, the proposed algorithm was verified by numerical experiments, and compared with the matrix block iterative inversion method. Results show that both methods had high computational efficiency and accuracy, but the computational burden of the proposed method was lower and the computational speed was faster, which verifies its accuracy and effectiveness. This algorithm can be used to analyze the Fisher information fusion process for every node or edge in arbitrary coupling undirected network navigation and positioning model.