Graphs serve as generic tools to encode the underlying relational structure of data. Often this graph is not given, and so the task of inferring it from nodal observations becomes important. Traditional approaches formulate a convex inverse problem with a smoothness promoting objective and rely on iterative methods to obtain a solution. In supervised settings where graph labels are available, one can unroll and truncate these iterations into a deep network that is trained end-to-end. Such a network is parameter efficient and inherits inductive bias from the optimization formulation, an appealing aspect for data constrained settings in, e.g., medicine, finance, and the natural sciences. But typically such settings care equally about uncertainty over edge predictions, not just point estimates. Here we introduce novel iterations with independently interpretable parameters, i.e., parameters whose values - independent of other parameters' settings - proportionally influence characteristics of the estimated graph, such as edge sparsity. After unrolling these iterations, prior knowledge over such graph characteristics shape prior distributions over these independently interpretable network parameters to yield a Bayesian neural network (BNN) capable of graph structure learning (GSL) from smooth signal observations. Fast execution and parameter efficiency allow for high-fidelity posterior approximation via Markov Chain Monte Carlo (MCMC) and thus uncertainty quantification on edge predictions. Synthetic and real data experiments corroborate this model's ability to provide well-calibrated estimates of uncertainty, in test cases that include unveiling economic sector modular structure from S$\&$P$500$ data and recovering pairwise digit similarities from MNIST images. Overall, this framework enables GSL in modest-scale applications where uncertainty on the data structure is paramount.

翻译：图是编码数据底层关系结构的通用工具。通常该图并非给定，因此从节点观测中推断图的任务变得至关重要。传统方法通过构建具有平滑性促进目标的凸逆问题，并依赖迭代方法获得解。在图标签可用的监督设置中，可将这些迭代展开并截断为端到端训练的深度网络。此类网络参数高效，并从优化公式中继承了归纳偏置，这对于医学、金融和自然科学等数据受限场景具有吸引力。但通常此类场景同样关注边预测的不确定性，而不仅仅是点估计。本文提出了具有独立可解释参数的新型迭代，即其值独立于其他参数设置、能按比例影响估计图特征（如边稀疏性）的参数。展开这些迭代后，关于此类图特征的先验知识形塑了这些独立可解释网络参数的先验分布，从而得到一个能够从平滑信号观测中进行图结构学习的贝叶斯神经网络。快速执行与参数效率使得通过马尔可夫链蒙特卡洛方法实现高保真后验近似成为可能，从而实现对边预测的不确定性量化。合成与真实数据实验验证了该模型在提供校准良好的不确定性估计方面的能力，测试案例包括从标普500数据揭示经济板块的模块化结构，以及从MNIST图像中恢复成对数字相似性。总体而言，该框架为数据结构不确定性至关重要的中等规模应用场景实现了图结构学习。