On Large-Scale System of Sparse Linear Equations with Tree Structures and its Applications to Fiber Optic Networks

Fiber optic networks require precise time synchronization, yet asymmetric edges introduce delays that challenge accurate correction. Current solutions like manual detection or the deployment of Global Navigation Satellite System (GNSS) receivers are impractical due to high costs and labor requirements. Statistically, delay detection is complicated by non-identifiability, fixed tree topologies, and absent structured sparsity assumptions. To overcome these difficulties, we propose a novel method that reformulates the original problem into solving a tree-structured system of linear equations (TSLE) with a regularization term to exploit the sparsity of asymmetric edges. The method is further enhanced through merging techniques to leverage the inherent tree structure. Additionally, a group partitioning algorithm is introduced to identify the Local Uncertainty Region (LUR), providing strategic guidance for GNSS receiver placement. Theoretically, we introduce the concept of equivalence, derive necessary and sufficient conditions for its existence, and examine the associated structural properties. We also theoretically assess the effectiveness of the LUR in reducing estimation uncertainty. Furthermore, we establish that the oracle estimator is asymptotically a local minimum of the penalized objective function. Empirical studies demonstrate that our proposed approach successfully identifies all asymmetric edges, offering a scalable and practical solution for real-world applications.