Random Gegenbauer Features for Scalable Kernel Methods
This work addresses scalability issues in kernel methods for machine learning practitioners, offering an incremental improvement in approximation efficiency.
The authors tackled the problem of approximating a broad class of kernel functions, including dot-product, Gaussian, and Neural Tangent kernels, by proposing random Gegenbauer features, which they proved have subspace embedding guarantees and empirically outperform recent methods.
We propose efficient random features for approximating a new and rich class of kernel functions that we refer to as Generalized Zonal Kernels (GZK). Our proposed GZK family, generalizes the zonal kernels (i.e., dot-product kernels on the unit sphere) by introducing radial factors in their Gegenbauer series expansion, and includes a wide range of ubiquitous kernel functions such as the entirety of dot-product kernels as well as the Gaussian and the recently introduced Neural Tangent kernels. Interestingly, by exploiting the reproducing property of the Gegenbauer polynomials, we can construct efficient random features for the GZK family based on randomly oriented Gegenbauer kernels. We prove subspace embedding guarantees for our Gegenbauer features which ensures that our features can be used for approximately solving learning problems such as kernel k-means clustering, kernel ridge regression, etc. Empirical results show that our proposed features outperform recent kernel approximation methods.