{"ID":5935715,"CreatedAt":"2026-07-07T01:22:02.77346169Z","UpdatedAt":"2026-07-07T02:10:06.972658124Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.03367","arxiv_id":"2607.03367","title":"Minimax Estimation of Kernel Stein Discrepancy: Trace versus Hilbert-Schmidt Scales","abstract":"Kernel Stein Discrepancy (KSD) compares a sample to a fixed target distribution known only through its score, and is widely used for goodness-of-fit testing, sample quality assessment, and approximate inference. We study the estimation of $\\operatorname{KSD}(P_0,P)$ from $n$ independent observations and identify the sharp spectral constant governing the minimax risk: it is the Hilbert-Schmidt norm of the Stein covariance operator $C_\\star$, giving the minimax scale $\\sqrt{\\|C_\\star\\|_{\\mathrm{HS}}/n}$. This scale is attained by the positive-part square-root U-statistic, whereas the standard plug-in V-statistic remains at the trace scale $\\sqrt{\\operatorname{tr}(C_\\star)/n}$ and is therefore suboptimal by the fourth root of the effective rank of $C_\\star$; for a Gaussian target with a fixed-bandwidth Gaussian kernel this factor is exponential in the dimension.","short_abstract":"Kernel Stein Discrepancy (KSD) compares a sample to a fixed target distribution known only through its score, and is widely used for goodness-of-fit testing, sample quality assessment, and approximate inference. We study the estimation of $\\operatorname{KSD}(P_0,P)$ from $n$ independent observations and identify the sh...","url_abs":"https://arxiv.org/abs/2607.03367","url_pdf":"https://arxiv.org/pdf/2607.03367v1","authors":"[\"Davit Gogolashvili\"]","published":"2026-07-03T14:22:58Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"stat.ML\"]","methods":"[]","has_code":false}