Regularization vs. Convolution in Kernel-Based Hypothesis Testing

The maximum mean discrepancy (MMD) test is a popular nonparametric test on general domains. Two lines of work have studied its optimality through structurally different approaches. The first, a convolution approach on Rᵈ, uses translation-invariant kernels with a shrinking bandwidth and is minimax optimal over Sobolev balls under L²-separation; the second, a regularization approach on general measurable spaces, uses spectral regularization of the kernel covariance operator and is minimax optimal over Hellinger- and χ²-separated alternatives. In this paper, we study the regularization approach over Sobolev balls under L²-separation, which reveals the connection between the two approaches. We show that, with a translation-invariant kernel on Rᵈ, both are ways of meeting a single condition on the kernel's spectral density: the spectral regularized discrepancy captures the L²-separation $Δ$ of a Sobolev alternative of smoothness s as soon as the regularization parameter $λ$ is below the spectral density on frequencies up to Δ-1/(2s). A kernel with a shrinking bandwidth h meets this condition with λ hᵈ, and under this coupling the constant regularizer, which gives the MMD test itself, attains the Sobolev minimax rate. This explains, from a spectral perspective, the need for a shrinking bandwidth in the convolution approach. With a non-constant spectral regularizer, our analysis requires the bandwidth to shrink only when the spectral density decays faster than any polynomial, as for the Gaussian kernel. Thus, for Matérn and Laplace kernels, a fixed bandwidth with λ→ 0 attains the Sobolev minimax rate. Finally, an adaptive test over a grid of bandwidths and regularization parameters is minimax optimal, up to a loglog factor, over both classes without prior knowledge of the smoothness.