We consider a specific graph learning task: reconstructing a symmetric matrix that represents an underlying graph using linear measurements. We present a sparsity characterization for distributions of random graphs (that are allowed to contain high-degree nodes), based on which we study fundamental tradeoffs between the number of measurements, the complexity of the graph class, and the probability of error. We first derive a necessary condition on the number of measurements. Then, by considering a three-stage recovery scheme, we give a sufficient condition for recovery. Furthermore, assuming the measurements are Gaussian IID, we prove upper and lower bounds on the (worst-case) sample complexity for both noisy and noiseless recovery. In the special cases of the uniform distribution on trees with n nodes and the Erdos-Rényi (n, p) class, the fundamental trade-offs are tight up to multiplicative factors with noiseless measurements. In addition, for practical applications, we design and implement a polynomialtime (in n) algorithm based on the three-stage recovery scheme. Experiments show that the heuristic algorithm outperforms basis pursuit on star graphs. We apply the heuristic algorithm to learn admittance matrices in electric grids. Simulations for several canonical graph classes and IEEE power system test cases demonstrate the effectiveness and robustness of the proposed algorithm for parameter reconstruction.

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从线性测量中学习图形：基本的取舍和应用

我们考虑一个特定的图学习任务：使用线性度量重建表示基础图的对称矩阵。我们为随机图（允许包含高级节点）的分布提供了稀疏性表征，在此基础上，我们研究了测量次数，图类的复杂性和错误概率之间的基本权衡。我们首先得出测量次数的必要条件。然后，通过考虑三阶段恢复方案，我们给出了充分的恢复条件。此外，假设测量是高斯IID，我们证明了在有噪和无噪恢复方面（最坏情况）样本复杂度的上限和下限。在具有n个节点和Erdos-Rényi（n，p）类的树上均匀分布的特殊情况下，在无噪声测量的情况下，基本的权衡取舍于乘数因子。另外，针对实际应用，我们基于三级恢复方案设计并实现了多项式时间（in n）算法。实验表明，启发式算法的性能优于星图的基本追求。我们应用启发式算法来学习电网中的导纳矩阵。对几种规范图类和IEEE电力系统测试案例的仿真证明了所提出算法用于参数重构的有效性和鲁棒性。实验表明，启发式算法的性能优于星图的基本追求。我们应用启发式算法来学习电网中的导纳矩阵。对几种规范图类和IEEE电力系统测试案例的仿真证明了所提出算法用于参数重构的有效性和鲁棒性。实验表明，启发式算法的性能优于星图的基本追求。我们应用启发式算法来学习电网中的导纳矩阵。对几种规范图类和IEEE电力系统测试案例的仿真证明了所提出算法用于参数重构的有效性和鲁棒性。