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Kernel ridge regression (KRR) is a standard method for performingnon-parametric regression over reproducing kernel Hilbert spaces. Given $n$samples, the time and space complexity of computing the KRR estimate scale as$\mathcal{O}(n^3)$ and $\mathcal{O}(n^2)$ respectively, and so is prohibitivein many cases. We propose approximations of KRR based on $m$-dimensionalrandomized sketches of the kernel matrix, and study how small the projectiondimension $m$ can be chosen while still preserving minimax optimality of theapproximate KRR estimate. For various classes of randomized sketches, includingthose based on Gaussian and randomized Hadamard matrices, we prove that itsuffices to choose the sketch dimension $m$ proportional to the statisticaldimension (modulo logarithmic factors). Thus, we obtain fast and minimaxoptimal approximations to the KRR estimate for non-parametric regression.

Randomized sketches for kernels: Fast and optimal non-parametric regression