Nyström Kernel Stein Discrepancy Tests

Over the past 20 years, kernel discrepancies have been leveraged as a highly powerful tool for quantifying the disagreement of distributions, with numerous successful applications in two-sample, goodness-of-fit, and independence testing, among others. Their fastest estimators are known to converge at a parametric rate---$n^{-1/2}$---under mild conditions. While this rate is known to be minimax optimal on $\mathbb R^d$ under strict assumptions with bounded kernels, little is known about its optimality beyond the finite-dimensional Euclidean setting with unbounded kernels. In this work, we prove that the minimax lower bound of estimation of the most popular kernel discrepancies (maximum mean discrepancy, Hilbert-Schmidt independence criterion and kernel Stein discrepancy; MMD, HSIC, KSD) is $n^{-1/2}$ on general topological spaces, and under mild assumptions on the kernel; the same rates are shown (as corollaries) to hold for the estimation of the mean embedding and the centered cross-covariance operator. Our results settle the question of optimal estimation of these kernel discrepancies.

Large-scale kernel ridge regression (KRR) is limited by the need to store a large kernel matrix K_t. To avoid storing the entire matrix K_t, Nystrom methods subsample a subset of columns of the kernel matrix, and efficiently find an approximate KRR solution on the reconstructed matrix. The chosen subsampling distribution in turn affects the statistical and computational tradeoffs. For KRR problems, recent works show that a sampling distribution proportional to the ridge leverage scores (RLSs) provides strong reconstruction guarantees for the approximation. While exact RLSs are as difficult to compute as a KRR solution, we may be able to approximate them well enough. In this paper, we study KRR problems in a sequential setting and introduce the INK-ESTIMATE algorithm, that incrementally computes the RLSs estimates. INK-ESTIMATE maintains a small sketch of K_t, that at each step is used to compute an intermediate estimate of the RLSs. First, our sketch update does not require access to previously seen columns, and therefore a single pass over the kernel matrix is sufficient. Second, the algorithm requires a fixed, small space budget to run dependent only on the effective dimension of the kernel matrix. Finally, our sketch provides strong approximation guarantees on the distance between the true kernel matrix and its approximation, and on the statistical risk of the approximate KRR solution at any time, because all our guarantees hold at any intermediate step.

Existing two-sample testing techniques, particularly those based on choosing a kernel for the Maximum Mean Discrepancy (MMD), often assume equal sample sizes from the two distributions. Applying these methods in practice can require discarding valuable data, unnecessarily reducing test power. We address this long-standing limitation by extending the theory of generalized U-statistics and applying it to the usual MMD estimator, resulting in new characterization of the asymptotic distributions of the MMD estimator with unequal sample sizes (particularly outside the proportional regimes required by previous partial results). This generalization also provides a new criterion for optimizing the power of an MMD test with unequal sample sizes. Our approach preserves all available data, enhancing test accuracy and applicability in realistic settings. Along the way, we give much cleaner characterizations of the variance of MMD estimators, revealing something that might be surprising to those in the area: while zero MMD implies a degenerate estimator, it is sometimes possible to have a degenerate estimator with nonzero MMD as well; we give a construction and a proof that it does not happen in common situations.