maximum mean discrepancy

Addressing the dual challenges of inflated Type I error rates (loss of test-level control) and low statistical power in conditional independence testing, this paper proposes a data-efficient kernel-based testing framework. The method employs kernel ridge regression and introduces, for the first time in this setting, three principled bias-correction strategies: data splitting, auxiliary data utilization, and restriction to simplified function classes—ensuring rigorous asymptotic and finite-sample control of the significance level. Theoretically, the approach guarantees convergence of the Type I error rate to the nominal significance level while enhancing detection power for complex dependency structures. Extensive experiments on diverse synthetic and real-world datasets demonstrate that the proposed method achieves precise Type I error control and substantially outperforms state-of-the-art competitors—including KCIT and RCIT—in statistical power, with improved robustness and reliability.

This paper addresses the goodness-of-fit testing problem under composite hypotheses: “Does a model belong to a given parametric family?” We propose a unified, data-splitting-free, kernel-based testing framework. For the first time, it enables parameter estimation and hypothesis testing to share the same dataset while rigorously controlling the Type-I error rate. The method accommodates unnormalized densities and simulator-based models without requiring explicit density evaluation. By integrating maximum mean discrepancy (MMD), kernel Stein discrepancy (KSD), and minimum distance estimation—within a theoretically grounded composite null testing framework—it substantially improves statistical power and broadens applicability. Experiments on unnormalized density models and biological cell-network simulators demonstrate its effectiveness. Theoretically, the test level is precisely controllable, and the framework extends the scope of goodness-of-fit testing to complex, intractable models.

Existing kernel-based goodness-of-fit (GoF) tests fail under both qualitative and quantitative robustness, and robustification strategies—such as tilted kernels—cannot simultaneously satisfy both criteria in GoF testing. Method: Addressing the practical question “Is the model sufficiently accurate?”, we propose the first robust GoF testing framework based on a kernel Stein discrepancy (KSD) ball. This framework rigorously formalizes robust GoF testing and theoretically establishes that conventional kernel tests—and their tilted-kernel variants—lack dual robustness. Contribution/Results: Our test achieves both stability and statistical power under diverse contamination models—including Huber contamination and density bands—enabling unified robust modeling. Empirical evaluation confirms its effectiveness in finite-sample settings, resolving a long-standing challenge in designing robust kernel-based GoF tests.

Traditional kernel Maximum Mean Discrepancy (MMD) two-sample tests rely on permutation to determine critical thresholds, ensuring finite-sample validity but incurring an O(n²) computational cost per permutation—prohibitively expensive for large samples. This paper proposes the cross-MMD test statistic: by splitting samples to construct a U-statistic, and combining studentization with a Gaussian kernel, it yields the first kernel MMD test that requires no permutations. The method achieves asymptotic normality with a single O(n²) computation, while preserving finite-sample validity, statistical consistency, and minimax optimal detection rates under local alternatives. Theoretically and empirically, cross-MMD accelerates testing by over an order of magnitude compared to permutation-based approaches on large samples, with only a marginal loss in power, and maintains strong consistency against any fixed distributional discrepancy.

To address the high computational cost, low statistical power, and bandwidth sensitivity of kernel two-sample tests on high-dimensional, large-scale data, this paper proposes a parameter-free robust kernel test. The method avoids bandwidth selection entirely while ensuring reliability and high power. Its core contributions are threefold: (1) a novel test statistic designed via theoretical analysis to eliminate power loss from data splitting; (2) a non-asymptotic significance control mechanism guaranteeing validity under finite samples; and (3) inherent suitability for high-dimensional settings, delivering uniformly high power across diverse alternative hypotheses. Experiments on synthetic and real-world datasets demonstrate that the proposed method achieves 10–100× speedup over MMD and state-of-the-art large-scale kernel tests, with average power gains of 15%–40%, all without any bandwidth tuning.

This work addresses the fundamental challenge of measuring discrepancies between conditional distributions in statistics and machine learning. The authors propose Conditional Maximum Mean Discrepancy (CMMD), a unified framework grounded in reproducing kernel Hilbert space embeddings, which establishes a hierarchical family ranging from CMMD₀ to CMMDₛ and elucidates their intrinsic mathematical relationships. A key innovation is the introduction of a doubly robust estimator that guarantees consistent estimation as long as either the conditional mean model or the weighting model is correctly specified. Combining operator smoothing with nonparametric techniques, the theoretical analysis is complemented by empirical results demonstrating that CMMD effectively captures complex conditional dependence structures and significantly outperforms existing methods in conditional distribution testing tasks.

Traditional goodness-of-fit tests struggle to distinguish between “no significant difference” and “practical equivalence,” as failure to reject the null hypothesis may merely reflect insufficient test power. This work proposes the first kernel-based framework for full-distribution equivalence testing, leveraging Kernel Stein Discrepancy (KSD) and Maximum Mean Discrepancy (MMD) to quantify the distance between distributions while incorporating a prespecified minimum equivalence margin. By employing asymptotic normal approximations and bootstrap procedures to compute critical values, the method overcomes the limitations of existing equivalence tests, which are typically confined to parametric models or specific moments. Numerical experiments demonstrate that the proposed approach reliably assesses whether two distributions are equivalent within the specified margin while effectively controlling both Type I and Type II error rates.

This work addresses the need for more efficient, robust, and flexible metrics for measuring distances between probability distributions in statistical inference and numerical integration. Centered on kernel methods, we propose an efficient estimator for Maximum Mean Discrepancy (MMD), develop novel MMD-based approaches for conditional expectation estimation and integral calibration, and introduce a new family of distance measures—kernel quantile discrepancies—that effectively overcome MMD’s limitations in tail sensitivity and discriminative power. Both theoretical analysis and empirical experiments demonstrate that the proposed methods offer strong scalability, computational efficiency, and superior performance, thereby providing more powerful and practical kernel-based tools for nonparametric statistics and integration tasks.

This study addresses the two-sample testing problem without distributional assumptions. It proposes PReLU-TST, a novel approach based on integral probability metrics (IPMs), which constructs a nonparametric test statistic using a parametrized discriminator consisting of only a single neuron. This design retains the flexibility of nonparametric methods while substantially improving computational efficiency. The proposed test is proven to be consistent and asymptotically equivalent to classical nonparametric IPM-based tests. Empirical evaluations demonstrate that PReLU-TST achieves higher or at least comparable finite-sample testing power against existing methods across a range of synthetic and real-world datasets.