NeuralEF: Deconstructing Kernels by Deep Neural Networks
This work addresses scalability and implementation issues in kernel eigenfunction approximation for researchers and practitioners in machine learning, offering a more efficient parametric method.
The paper tackles the problem of learning principal eigenfunctions of kernel integral operators, which is crucial for many ML tasks, by introducing a new objective function that generalizes EigenGame to function space, enabling accurate approximations of various kernels and scaling up linearised Laplace approximation for deep neural networks on image classification datasets.
Learning the principal eigenfunctions of an integral operator defined by a kernel and a data distribution is at the core of many machine learning problems. Traditional nonparametric solutions based on the Nystr{ö}m formula suffer from scalability issues. Recent work has resorted to a parametric approach, i.e., training neural networks to approximate the eigenfunctions. However, the existing method relies on an expensive orthogonalization step and is difficult to implement. We show that these problems can be fixed by using a new series of objective functions that generalizes the EigenGame~\citep{gemp2020eigengame} to function space. We test our method on a variety of supervised and unsupervised learning problems and show it provides accurate approximations to the eigenfunctions of polynomial, radial basis, neural network Gaussian process, and neural tangent kernels. Finally, we demonstrate our method can scale up linearised Laplace approximation of deep neural networks to modern image classification datasets through approximating the Gauss-Newton matrix. Code is available at \url{https://github.com/thudzj/neuraleigenfunction}.