A new method optimizes MMD test power by dynamically selecting kernels, overcoming traditional trade-offs.
problem Fixed kernels fail to distinguish certain distributions, leading to overfitting and variance collapse.
method Complexity-Penalized MMD (CP-MMD) criterion, derived from concentration inequality, optimizes kernel selection.
result CP-MMD maximizes true test power while ensuring unconditional Type-I validity, matching or exceeding state-of-the-art performance.
A new method uses neural tangent kernel to efficiently compute MMD statistic.
problem Efficiently computing Maximum Mean Discrepancy (MMD) statistic with low memory and computational complexity.
method Identifies a connection between neural tangent kernel (NTK) and MMD to develop a computationally and memory-efficient approach.
result The proposed NTK-MMD statistic is validated through numerical experiments on synthetic and real-world datasets.
A new MMD-based test combines kernels for two-sample testing without splitting data.
problem Efficiently testing if two datasets come from the same distribution without splitting data.
method Proposes a novel statistic based on Maximum Mean Discrepancy (MMD) that combines kernels, proving concentration bounds and showing data-dependent kernel selection.
result Exponential concentration bounds and improved test power compared to existing methods.
A new test statistic speeds up MMD while maintaining power.
problem Efficiently testing two distributions without permutations.
method Cross-MMD statistic based on sample-splitting and studentization.
result Cross-MMD has a limiting standard Gaussian distribution under the null.
EVI-MMD approximates target distributions via MMD minimization with adaptive kernel.
problem Approximating target distributions using kernel discrepancy methods.
method EVI-MMD uses Maximum Mean Discrepancy (MMD) to minimize kernel discrepancy, solving ODEs with implicit Euler scheme and L-BFGS optimization.
result EVI-MMD with adaptive bandwidth selection significantly improves performance in sampling problems.
Paper improves MMD flow efficiency with Riesz kernels for image generation.
problem High computational costs in MMD flows for large scale computations.
method Introduces Riesz kernels and sliced MMD for efficient computation.
result Efficient computation of MMD gradients in one-dimensional setting.
A test for comparing function samples using MMD.
problem Testing if two functional data samples come from the same distribution.
method Maximum Mean Discrepancy (MMD) for functional data, with theoretical scaling analysis.
result The proposed test is effective and robust to functional reconstructions.
Two-sample tests using MMD control type I error and achieve optimal power.
problem Developing reliable nonparametric two-sample tests for small sample sizes.
method Maximum Mean Discrepancy (MMD) for constructing novel nonparametric tests, proving non-asymptotic error control and optimality.
result MMDAgg test controls type I error and achieves minimax rate over Sobolev balls, outperforming other tests.
A new kernel test reduces noise in MMD by focusing on leading eigen-directions.
problem Noise in trailing directional components degrades power of standard kernel two-sample tests.
method Truncate MMD spectral decomposition, retaining only leading eigen-directions.
result Our method achieves superior power and robustness, especially in high-dimensional and unbalanced settings.
sig-MMD tests compare path distributions using kernel methods.
problem Comparing path distributions in stochastic processes.
method Signature kernel for path space valued distributions.
result sig-MMD can lead to Type 2 errors in limited data settings.
Unified framework for optimal kernel tests across MMD, HSIC, and KSD.
problem Optimal testing in kernel-based hypothesis testing frameworks.
method Unified derivation of minimax rates, adaptive kernel selection methods.
result Unified power results across MMD, HSIC, and KSD.
This thesis improves kernel-based distances for statistical inference and integration.
problem Efficiently measuring distances between probability distributions for robust and smooth modeling.
method Kernel-based distances, focusing on maximum mean discrepancy (MMD) and novel kernel quantile discrepancies.
result Improved MMD estimators for simulation-based inference and conditional expectations.
The article introduces practical estimators for kernel discrepancies.
problem Estimating kernel discrepancies accurately and efficiently.
method Presented various estimators for MMD, HSIC, and KSD, including V-statistics, U-statistics, and incomplete U-statistics. Stressed the importance of kernel bandwidth and introduced adaptive estimators.
result Adaptive estimators combining multiple estimators with various kernels address the problem of kernel selection.
MMD test detects adversarial attacks by addressing kernel limitations and non-independence issues.
problem MMD test's failure to detect adversarial attacks.
method Replaced Gaussian kernel with deep kernel, maximized test power, and used wild bootstrap for non-independence.
result MMD test is aware of adversarial attacks.
We investigate the training and performance of generative adversarial networks using the Maximum Mean Discrepancy (MMD) as critic, termed MMD GANs. As our main theoretical contribution, we clarify the situation with bias in GAN loss functions raised by recent work: we show that gradient estimators used in the optimizat…
Proposes a new method for posterior sampling using MMD with negative distance kernel.
problem Posterior sampling and conditional generative modeling.
method Approximates joint distribution using discrete Wasserstein gradient flows of MMD with negative distance kernel.
result Establishes an error bound for posterior distributions and proves the method is a Wasserstein gradient flow.
Gradient descent on MMD GAN parameter space converges globally to target distribution.
problem Convergence of gradient descent in Maximum Mean Discrepancy (MMD) GANs.
method Proposes a parametric kernelized gradient flow that mimics the min-max game in gradient regularized MMD GAN.
result Gradient descent on the generator's parameter space in gradient regularized MMD GAN is globally convergent to the target distribution under certain conditions.
New method finds points for approximating distributions faster.
problem Approximating target probability distributions using finite points.
method Stationary MMD points computed via MMD gradient flows.
result Stationary MMD points converge faster than global minimizers.
New KQEs improve probability metrics without mean function constraints.
problem Improving probability metrics without relying on mean function representations.
method Kernel quantile embeddings (KQEs) to construct new distances.
result KQEs offer a competitive alternative to MMD with near-linear cost.
Regularizes f-divergences with MMD to analyze Wasserstein flows.
problem Limitations of f-divergences in measures' support. method Rewriting MMD regularization as Moreau envelope in RKHS, analyzing gradients.
result Analysis of Wasserstein flows of MMD-regularized f-divergences. Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. T…
New kernel improves MMDs with theoretical guarantees for gradient flows.
problem Non-smoothness of negative distance kernel in MMDs.
method Smoothed 1D absolute value function followed by fractional integral transform.
result Improved theoretical guarantees for Wasserstein gradient flows.
Unified method for MMD variance estimation improves accuracy and computational efficiency.
problem Variance estimation for MMD in nonparametric testing.
method Unified finite-sample characterization of MMD variance through U-statistic and Hoeffding decomposition; exact acceleration method for univariate case.
result Unified estimators improve accuracy and computational efficiency for MMD variance.
Kernel-based mean-field games use MMD penalties for interaction and target costs.
problem Optimizing mean-field games with specific cost functions.
method Kernel structure, random Fourier U-statistics, neural network training.
result Sample-level convergence theorem and rate of convergence proved.
Study improves MMD estimation for two distributions with mismeasured data.
problem Estimating MMD under contaminated data.
method Partial identification of MMD bounds with a fast convergence rate.
result Proposes a method to estimate MMD bounds that converge faster than alternatives.
MMD-B-Fair learns fair representations by minimizing MMD test power.
problem Learning fair representations of data while preserving target attributes.
method Kernel two-sample testing and block testing schemes.
result Minimizing MMD test power allows hiding sensitive attribute information.
Kernels are powerful and versatile tools in machine learning and statistics. Although the notion of universal kernels and characteristic kernels has been studied, kernel selection still greatly influences the empirical performance. While learning the kernel in a data driven way has been investigated, in this paper we e…
Boosts kernel two-sample test power with multiple kernels.
problem Detecting differences between two distributions over metric spaces.
method Combining MMD estimates over multiple kernels using Mahalanobis distance.
result More powerful in detecting a wide range of alternatives in finite samples.
A new measure scales MMD to assess distribution closeness.
problem Testing statistical significance of distribution closeness.
method Norm-adaptive MMD (NAMMD) for distributional discrepancy.
result NAMMD-based DCT has higher test power than MMD-based DCT.
Generative moment matching network (GMMN) is a deep generative model that differs from Generative Adversarial Network (GAN) by replacing the discriminator in GAN with a two-sample test based on kernel maximum mean discrepancy (MMD). Although some theoretical guarantees of MMD have been studied, the empirical performanc…
Study on kernel tests for high-dimensional data, focusing on MMD and CLT.
problem Asymptotic behavior of kernel two-sample tests in high dimensions and large samples.
method Maximum mean discrepancy (MMD) with isotropic kernels, deriving asymptotic expansions and CLT.
result Interplay between moment discrepancy and dimension-and-sample orders in kernel tests.
The paper shows how MMD metrizes weak convergence for certain kernels.
problem Characterizing MMD metrizing weak convergence for a wide class of kernels.
method Proving MMD metrizes weak convergence for specific kernels on a locally compact space.
result Corrected prior results and identified new kernels metrizing weak convergence.
New method for MMD with unequal sample sizes improves test power.
problem Existing MMD methods assume equal sample sizes, discarding valuable data.
method Extended generalized U-statistics to handle unequal sample sizes.
result New asymptotic distributions and power optimization for MMD with unequal sample sizes.
New metrics improve probabilistic forecasting, especially for rare events.
problem Current evaluation frameworks for probabilistic forecasting assume independence and lack sensitivity to tail events.
method Proposed signature kernel-based metrics: Sig-MMD and CSig-MMD.
result These metrics capture complex dependencies and prioritize tail event prediction.
The maximum mean discrepancy (MMD) is a recently proposed test statistic for two-sample test. Its quadratic time complexity, however, greatly hampers its availability to large-scale applications. To accelerate the MMD calculation, in this study we propose an efficient method called FastMMD. The core idea of FastMMD is …
This note optimizes distributions using kernel mean embeddings with a new parameterization.
problem Optimizing distributions using kernel mean embeddings is challenging due to the difficulty of characterizing probability distribution vectors.
method Proposes a new parameterization of positive functions using kernel sums-of-squares to fit distributions in the MMD geometry.
result Distributions with kernel sum-of-squares densities are dense in the MMD geometry, allowing optimization in the finite-sample setting.
The Maximum Mean Discrepancy (MMD) has found numerous applications in statistics and machine learning, most recently as a penalty in the Wasserstein Auto-Encoder (WAE). In this paper we compute closed-form expressions for estimating the Gaussian kernel based MMD between a given distribution and the standard multivariat…
Distributionally robust optimization (DRO) has attracted attention in machine learning due to its connections to regularization, generalization, and robustness. Existing work has considered uncertainty sets based on phi-divergences and Wasserstein distances, each of which have drawbacks. In this paper, we study DRO wit…
New conditions ensure MMDs separate and converge to target distributions.
problem Ensuring MMDs separate and converge to target distributions.
method Deriving new sufficient and necessary conditions for MMDs on separable metric spaces.
result First KSDs that exactly metrize weak convergence to P.
Improves probability distribution compression with KT algorithm.
problem Efficiently compressing probability distributions.
method Kernel thinning (KT) algorithm with four improvements.
result KT yields tighter, dimension-free guarantees for any kernel.
A new distance metric compares probability distributions using kernel covariance operators.
problem Comparing probability distributions in machine learning tasks.
method Introduces a novel distance metric based on Schatten norm of kernel covariance operators.
result The new distance metric is more discriminative and robust to hyperparameters.
We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics literature; on the other, maximum mean discrepancies (MMD), that is, distances between embeddings of distributions to reproduc…
New method approximates MMD using pseudo-differential operators and singular values.
problem Approximating MMD with pseudo-differential operators and singular values.
method Corresponding pseudo-differential operators to Mercer kernels, approximating p(x,y) with its first r singular values. result The new MMD distance measures the difference of two distributions with respect to r∗ local moments, where r∗ depends on singular values decay rate. The paper analyzes greedy algorithms for MMD minimization, showing their efficiency and approximation error.
problem Minimizing Maximum Mean Discrepancy (MMD) for probability measure quantization.
method Iterative algorithms including kernel herding, greedy MMD minimization, and Sequential Bayesian Quadrature (SBQ).
result The greedy algorithms have a lower approximation error than SBQ, but are significantly faster.
We characterize the asymptotic performance of nonparametric goodness of fit testing. The exponential decay rate of the type-II error probability is used as the asymptotic performance metric, and a test is optimal if it achieves the maximum rate subject to a constant level constraint on the type-I error probability. We …
Improved MMD test for non-Euclidean data with spectral regularization.
problem Inefficient and impractical MMD goodness-of-fit tests for non-Euclidean data.
method Spectral regularization of MMD test, extending results to general cases.
result Minimax optimal test for non-Euclidean data with appropriate regularization.
We study the maximum mean discrepancy (MMD) in the context of critical transitions modelled by fast-slow stochastic dynamical systems. We establish a new link between the dynamical theory of critical transitions with the statistical aspects of the MMD. In particular, we show that a formal approximation of the MMD near …
AGMMNs improve learning of copula models by adaptively selecting kernels.
problem Learning dependence structures in copula models.
method Adaptive bandwidth selection for MMD in GMMNs, increasing kernels based on validation loss.
result AGMMNs significantly improve training performance over GMMNs and parametric models.