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[2510.15058] The Minimax Lower Bound of Kernel Stein Discrepancy Estimation
Summary
This paper establishes the minimax lower bound of Kernel Stein Discrepancy (KSD) estimation, demonstrating its optimality and implications for high-dimensional data analysis.
Why It Matters
Understanding the minimax lower bound of KSD estimation is crucial for researchers in machine learning and statistics, as it provides a benchmark for the performance of KSD estimators. This work highlights the challenges posed by high-dimensional spaces, which is increasingly relevant in modern data science applications.
Key Takeaways
- The minimax lower bound for KSD estimation is established as n^{-1/2}.
- KSD estimators achieve optimal convergence rates, specifically √n.
- The difficulty of KSD estimation increases exponentially with dimensionality.
- Two distinct proof strategies are provided to support the findings.
- The results have significant implications for goodness-of-fit assessments in high-dimensional data.
Statistics > Machine Learning arXiv:2510.15058 (stat) [Submitted on 16 Oct 2025 (v1), last revised 20 Feb 2026 (this version, v2)] Title:The Minimax Lower Bound of Kernel Stein Discrepancy Estimation Authors:Jose Cribeiro-Ramallo, Agnideep Aich, Florian Kalinke, Ashit Baran Aich, Zoltán Szabó View a PDF of the paper titled The Minimax Lower Bound of Kernel Stein Discrepancy Estimation, by Jose Cribeiro-Ramallo and 4 other authors View PDF HTML (experimental) Abstract:Kernel Stein discrepancies (KSDs) have emerged as a powerful tool for quantifying goodness-of-fit over the last decade, featuring numerous successful applications. To the best of our knowledge, all existing KSD estimators with known rate achieve $\sqrt n$-convergence. In this work, we present two complementary results (with different proof strategies), establishing that the minimax lower bound of KSD estimation is $n^{-1/2}$ and settling the optimality of these estimators. Our first result focuses on KSD estimation on $\mathbb R^d$ with the Langevin-Stein operator; our explicit constant for the Gaussian kernel indicates that the difficulty of KSD estimation may increase exponentially with the dimensionality $d$. Our second result settles the minimax lower bound for KSD estimation on general domains. Comments: Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST) MSC classes: 62C20 (Primary) 46E22, 62B10 (Secondary) ACM classes: G.3; H.1.1; I.2.6 Cite as: arXiv:2510.15058 [...
