Historically, the machine learning community has derived spectral decompositions from graph-based approaches. We break with this approach and prove the statistical and computational superiority of the Galerkin method, which consists in restricting the study to a small set of test functions. In particular, we introduce implementation tricks to deal with differential operators in large dimensions with structured kernels. Finally, we extend on the core principles beyond our approach to apply them to non-linear spaces of functions, such as the ones parameterized by deep neural networks, through loss-based optimization procedures.

翻译：历史上，机器学习领域通常基于图方法推导谱分解。我们突破这一传统，证明了伽辽金法在统计和计算上的优越性——该方法通过将研究约束在一小组测试函数上实现。具体而言，我们引入实现技巧以处理带有结构化核的大规模微分算子。最后，我们将方法的核心原理拓展至非线性函数空间（例如由深度神经网络参数化的函数空间），并通过基于损失的优化过程进行应用。