This paper tackles the challenging problem of jointly inferring time-varying network topologies and imputing missing data from partially observed graph signals. We propose a unified non-convex optimization framework to simultaneously recover a sequence of graph Laplacian matrices while reconstructing the unobserved signal entries. Unlike conventional decoupled methods, our integrated approach facilitates a bidirectional flow of information between the graph and signal domains, yielding superior robustness, particularly in high missing-data regimes. To capture realistic network dynamics, we introduce a fused-lasso type regularizer on the sequence of Laplacians. This penalty promotes temporal smoothness by penalizing large successive changes, thereby preventing spurious variations induced by noise while still permitting gradual topological evolution. For solving the joint optimization problem, we develop an efficient Alternating Direction Method of Multipliers (ADMM) algorithm, which leverages the problem's structure to yield closed-form solutions for both the graph and signal subproblems. This design ensures scalability to large-scale networks and long time horizons. On the theoretical front, despite the inherent non-convexity, we establish a convergence guarantee, proving that the proposed ADMM scheme converges to a stationary point. Furthermore, we derive non-asymptotic statistical guarantees, providing high-probability error bounds for the graph estimator as a function of sample size, signal smoothness, and the intrinsic temporal variability of the graph. Extensive numerical experiments validate the approach, demonstrating that it significantly outperforms state-of-the-art baselines in both convergence speed and the joint accuracy of graph learning and signal recovery.

翻译：本文研究了从部分观测的图信号中联合推断时变网络拓扑并填补缺失数据这一具有挑战性的问题。我们提出了一个统一的非凸优化框架，旨在同时恢复一系列图拉普拉斯矩阵并重建未观测到的信号条目。与传统的解耦方法不同，我们的集成方法促进了图域与信号域之间的双向信息流动，从而获得了更强的鲁棒性，尤其是在高缺失数据情况下。为了捕捉真实的网络动态，我们在拉普拉斯矩阵序列上引入了融合套索型正则化项。该惩罚项通过惩罚相邻时间步之间的大幅变化来促进时间平滑性，从而防止噪声引起的虚假波动，同时仍允许拓扑结构发生渐进演化。针对这一联合优化问题，我们开发了一种高效的交替方向乘子法（ADMM）算法，该算法利用问题的结构，为图子问题和信号子问题均提供了闭式解。这一设计确保了算法可扩展至大规模网络和长时间序列。在理论方面，尽管问题本身具有非凸性，我们仍建立了收敛性保证，证明了所提出的ADMM方案会收敛到一个稳定点。此外，我们推导了非渐近统计保证，为图估计器提供了以样本量、信号平滑度及图本身的时间变异性为函数的高概率误差界。大量的数值实验验证了该方法的有效性，结果表明其在收敛速度以及图学习与信号恢复的联合精度方面均显著优于现有先进基线方法。