Graph is a highly generic and diverse representation, suitable for almost any data processing problem. Spectral graph theory has been shown to provide powerful algorithms, backed by solid linear algebra theory. It thus can be extremely instrumental to design deep network building blocks with spectral graph characteristics. For instance, such a network allows the design of optimal graphs for certain tasks or obtaining a canonical orthogonal low-dimensional embedding of the data. Recent attempts to solve this problem were based on minimizing Rayleigh-quotient type losses. We propose a different approach of directly learning the eigensapce. A severe problem of the direct approach, applied in batch-learning, is the inconsistent mapping of features to eigenspace coordinates in different batches. We analyze the degrees of freedom of learning this task using batches and propose a stable alignment mechanism that can work both with batch changes and with graph-metric changes. We show that our learnt spectral embedding is better in terms of NMI, ACC, Grassman distance, orthogonality and classification accuracy, compared to SOTA. In addition, the learning is more stable.

翻译：图是一种高度通用且多样化的表示形式，适用于几乎所有的数据处理问题。谱图理论已被证明能够提供强大的算法，并得到坚实的线性代数理论支持。因此，设计具有谱图特性的深度网络构建模块极具价值。例如，此类网络可为特定任务设计最优图，或获取数据的标准正交低维嵌入。近期解决该问题的尝试主要基于最小化瑞利商类型损失函数。我们提出了一种不同的方法——直接学习特征空间。直接方法在批量学习中面临一个严重问题：不同批次中特征到特征空间坐标的映射不一致。我们分析了在批量学习中学习该任务的自由度，并提出了一种稳定的对齐机制，该机制既能适应批次变化，也能适应图度量变化。实验表明，与现有最优方法相比，我们学习的谱嵌入在归一化互信息（NMI）、准确率（ACC）、格拉斯曼距离、正交性及分类准确率方面更优，且学习过程更加稳定。