Graph learning from signals is a core task in Graph Signal Processing (GSP). One of the most commonly used models to learn graphs from stationary signals is SpecT. However, its practical formulation rSpecT is known to be sensitive to hyperparameter selection and, even worse, to suffer from infeasibility. In this paper, we give the first condition that guarantees the infeasibility of rSpecT and design a novel model (LogSpecT) and its practical formulation (rLogSpecT) to overcome this issue. Contrary to rSpecT, the novel practical model rLogSpecT is always feasible. Furthermore, we provide recovery guarantees of rLogSpecT, which are derived from modern optimization tools related to epi-convergence. These tools could be of independent interest and significant for various learning problems. To demonstrate the advantages of rLogSpecT in practice, a highly efficient algorithm based on the linearized alternating direction method of multipliers (L-ADMM) is proposed. The subproblems of L-ADMM admit closed-form solutions and the convergence is guaranteed. Extensive numerical results on both synthetic and real networks corroborate the stability and superiority of our proposed methods, underscoring their potential for various graph learning applications.

翻译：从信号中学习图是图信号处理（GSP）中的核心任务。从平稳信号中学习图最常用的模型之一是SpecT。然而，其实用化形式rSpecT已知对超参数选择敏感，更严重的是，存在不可行性问题。本文首次给出了保证rSpecT不可行性的条件，并设计了一种新模型（LogSpecT）及其实用化形式（rLogSpecT）以克服该问题。与rSpecT相反，新型实用模型rLogSpecT始终可行。此外，我们提供了rLogSpecT的恢复保证，该保证源自与epi-收敛相关的现代优化工具。这些工具可能具有独立研究价值，并对各类学习问题意义重大。为展示rLogSpecT的实践优势，提出了一种基于线性化交替方向乘子法（L-ADMM）的高效算法。L-ADMM的子问题具有闭式解且收敛性得到保证。在合成网络与真实网络上的大量数值结果证实了我们所提出方法的稳定性与优越性，凸显了其在各类图学习应用中的潜力。