Regularization vs. Convolution in Kernel-Based Hypothesis Testing
Abstract
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 λasymp 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.
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BibTeX
@misc{hagrass2026regularization,
title = {{Regularization vs. Convolution in Kernel-Based Hypothesis Testing}},
author = {Omar Hagrass and Bharath K. Sriperumbudur},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06778},
eprint = {2611.06778},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06778},
}
RIS
TY - UNPB
TI - Regularization vs. Convolution in Kernel-Based Hypothesis Testing
AU - Omar Hagrass
AU - Bharath K. Sriperumbudur
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06778
UR - https://arxiv.org/abs/2611.06778
AB - 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 ℝ^d, uses translation-invariant kernels with a shrinking bandwidth and is minimax optimal over Sobolev balls under L^2-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 χ^2-separated alternatives. In this paper, we study the regularization approach over Sobolev balls under L^2-separation, which reveals the connection between the two approaches. We show that, with a translation-invariant kernel on ℝ^d, both are ways of meeting a single condition on the kernel's spectral density: the spectral regularized discrepancy captures the L^2-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 λasymp h^d, 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.
ER -