The Galerkin method beats Graph-Based Approaches for Spectral Algorithms
This work addresses a foundational problem in machine learning for researchers and practitioners by introducing a novel paradigm shift in spectral algorithms.
The authors tackled the problem of spectral decompositions in machine learning by proving the Galerkin method outperforms traditional graph-based approaches in statistical and computational aspects, with implementation tricks for handling differential operators in high dimensions.
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.