Distributed Adaptive Sampling for Kernel Matrix Approximation

Abstract

Most kernel-based methods, such as kernel or Gaussian process regression, kernel PCA, ICA, or $k$-means clustering, do not scale to large datasets, because constructing and storing the kernel matrix $\mathbf{K}_n$ requires at least $\mathcal{O}(n^2)$ time and space for $n$ samples. Recent works show that sampling points with replacement according to their ridge leverage scores (RLS) generates small dictionaries of relevant points with strong spectral approximation guarantees for $\mathbf{K}_n$. The drawback of RLS-based methods is that computing exact RLS requires constructing and storing the whole kernel matrix. In this paper, we introduce SQUEAK, a new algorithm for kernel approximation based on RLS sampling that sequentially processes the dataset, storing a dictionary which creates accurate kernel matrix approximations with a number of points that only depends on the effective dimension $d_{eff}(\gamma)$ of the dataset. Moreover since all the RLS estimations are efficiently performed using only the small dictionary, SQUEAK is the first RLS sampling algorithm that never constructs the whole matrix $\mathbf{K}_n$, runs in linear time $\widetilde{\mathcal{O}}(nd_{eff}(\gamma)^3)$ w.r.t. $n$, and requires only a single pass over the dataset. We also propose a parallel and distributed version of SQUEAK that linearly scales across multiple machines, achieving similar accuracy in as little as $\widetilde{\mathcal{O}}(\log(n)d_{eff}(\gamma)^3)$ time.