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.
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.
Paper develops metrics for random dynamical systems using vector-valued RKHSs.
problem Creating metrics for random nonlinear dynamical systems.
method Develops metrics on random dynamical systems using Perron-Frobenius operators in vector-valued reproducing kernel Hilbert spaces (vvRKHSs). Uses operator-valued kernels and time-wise independence criteria.
result Extends existing metrics for deterministic systems and introduces kernel maximal mean discrepancy for random processes.
Optimal tests for goodness of fit and two-sample problems using MMD and KSD.
problem Asymptotically optimal tests for goodness of fit and two-sample problems.
method Maximum Mean Discrepancy (MMD) and Kernel Stein Discrepancy (KSD) based tests.
result Optimal tests achieve the maximum exponential decay rate under specific conditions.
Optimizes kernel discrepancies by selecting subsets efficiently.
problem Improving kernel discrepancies for QMC methods.
method Introduces a novel subset selection algorithm for kernel discrepancies.
result Efficiently generates low-discrepancy samples from various distributions.
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.
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.
Study uses robust signature moments to characterize laws of stochastic processes.
problem Characterize laws of stochastic processes.
method Robust signature moments and maximum mean discrepancy metric.
result Derive a metric for laws of stochastic processes and kernelize it.
This work proves the optimal estimation rates for popular kernel discrepancies.
problem Estimating the disagreement of distributions using kernel discrepancies.
method Proving minimax lower bounds for MMD, HSIC, and KSD.
result The minimax lower bound for estimation of MMD, HSIC, and KSD is \( n^{-1/2} \) on general topological spaces.
New KCC-SDs improve distribution comparison in high dimensions.
problem Challenges in high-dimensional Stein discrepancies.
method Kernelized complete conditional Stein discrepancies (KCC-SDs).
result KCC-SDs outperform baselines in distinguishing distributions.
Sliced kernelized Stein discrepancy improves goodness-of-fit tests and model learning in high dimensions.
problem The curse-of-dimensionality in kernelized Stein discrepancy (KSD).
method Sliced Stein discrepancy and its scalable variants using optimal one-dimensional projections.
result Significantly outperforms KSD and baselines in goodness-of-fit tests and improves model learning.
Gradient flow for MMD converges to optimal solution, with regularization.
problem Optimizing neural networks using MMD as a metric.
method Constructing a Wasserstein gradient flow of MMD and studying its convergence properties.
result Gradient flow converges to global optimum under certain conditions.
This work proposes a new method to match distributions across different spaces using cycle-consistent maps.
problem Matching distributions across different spaces with consistent bidirectional maps.
method A novel unbalanced Monge optimal transport formulation for matching distributions on different spaces, employing cycle-consistent maps.
result The proposed discrepancy captures the cycle-consistent GAN framework and provides theoretical support.
PolyGraph Discrepancy improves graph generative model evaluation.
problem Inability of existing metrics to provide an absolute performance measure and comparability across different graph descriptors.
method Approximates Jensen-Shannon distance using binary classifiers trained to distinguish between real and generated graphs.
result PGD provides a more robust and insightful evaluation compared to MMD metrics.
We approximate the Sobolev discrepancy for finite dimensional kernels from samples.
problem Estimating the Sobolev discrepancy for complex models from finite data.
method Approximating the Sobolev discrepancy using finite samples and analyzing the approximation error.
result The Sobolev discrepancy can be approximated from finite samples and its error depends on the approximation and statistical errors.
Study finds polynomial convergence rate for Farey sequences linked to Riemann hypothesis.
problem Understanding convergence rates of maximum mean discrepancies for Farey sequences.
method Identifying positive-semidefinite kernels and their polynomial convergence rates.
result Polynomial convergence rate of maximum mean discrepancies of Farey sequences is equivalent to the Riemann hypothesis.
A new framework improves kernel Stein discrepancy tests for validating distributions.
problem Improving goodness-of-fit testing for non-normal distributions.
method Introducing Sf-KSD, a unifying framework for studying Stein operators in KSD-based tests.
result Sf-KSD guides the development of new tests and outperforms existing methods.
Deep neural networks can approximate any target probability distribution given certain conditions.
problem Approximating complex probability distributions with deep neural networks.
method Proving the existence of a deep neural network mapping that approximates a target distribution under various integral probability metrics.
result Upper bounds on the size of the neural network in terms of dimension and approximation error for different metrics.
Paper defines class discrepancy for machine learning problems and provides coresets.
problem Addressing discrepancies in machine learning models.
method Defines class discrepancy, provides techniques for bounding discrepancy, and develops coresets and streaming sketches.
result Establishes coresets of size O(sqrt{d}/epsilon) for various machine learning problems.
Framework synthesizes geological images minimizing patch distribution discrepancy.
problem Synthesizing realistic geological images from a single exemplar.
method Uses kernel discrepancies and generative neural networks for efficient synthesis.
result Synthesized images match visual patterns and spatial statistics of the exemplar.
Generative model for realistic text generation using feature matching.
problem Applying GAN to text generation due to convergence issues and difficulties with discrete data.
method Employing a long short-term memory network as generator and a convolutional network as discriminator. Matching latent feature distributions via a kernelized discrepancy metric.
result Superior performance in generating realistic-looking sentences compared to standard GAN.
Paper develops a minimax optimal test for goodness-of-fit using kernel Stein discrepancy.
problem Developing a robust goodness-of-fit test for general domains.
method Kernel Stein Discrepancy (KSD) with spectral regularization and adaptive testing.
result Proposed regularized test achieves minimax optimality up to a logarithmic factor.
A new metric compares true and learned causal graphs considering data and graph structure.
problem Comparing true and learned causal graphs accurately.
method Continuous Structural Intervention Distance (CSID) using conditional mean embeddings and maximum mean discrepancy.
result Validated the CSID with synthetic data, showing its effectiveness in comparing causal graphs.
A new method for kernel tests without data splitting increases power.
problem Lack of power in kernel-based tests due to data splitting.
method Selective inference framework to learn hyperparameters and test on full sample.
result Empirically larger test power without data splitting, regardless of split proportion.
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.
Paper develops a unified framework for measuring differences between conditional distributions.
problem Comparing conditional distributions in a unified and theoretically sound manner.
method Kernel embeddings and conditional maximum mean discrepancy (CMMD) framework.
result Established a coherent framework for measuring divergence between conditional distributions.
Improved KSD test for better detection of differences in distributions.
problem Low power of KSD test when distributions have same modes but different mixing proportions.
method Perturb the observed sample using Markov transition kernels to improve KSD test power.
result Perturbed KSD test can lead to substantially higher power than the original KSD test.
New findings show fixed-kernel discriminators are weaker than feature-learning ones.
problem Comparing performance of fixed-kernel and feature-learning discriminators.
method Using function classes F 2 \mathcal{F}_2 F 2 and F 1 \mathcal{F}_1 F 1 , constructing pairs of distributions, and linking IPMs with sliced Wasserstein distances. result Fixed-kernel IPM and SD cannot discriminate certain distributions that feature-learning IPM and SD can.
A new method for distribution regression using sliced Wasserstein distance.
problem Learning functions over spaces of probabilities.
method Proposes an OT-based estimator using the Sliced Wasserstein distance.
result Proves universal consistency and excess risk bounds for the proposed estimator.
Improves two-sample hypothesis testing using kernel divergences and scoring rules.
problem Two-sample hypothesis testing in machine learning.
method Proposes Kernel Scoring Rules and Divergences, including the Maximum Mean Discrepancy.
result Kernel Score provides more information about embedded distributions than Maximum Mean Discrepancy.
Kernel tests assess equivalence between distributions without assuming specific moments.
problem Traditional goodness-of-fit tests fail to detect meaningful distributional differences.
method Proposes kernel-based tests using kernel Stein discrepancy and Maximum Mean Discrepancy.
result Tests assess the absence of meaningful distributional differences under controlled error rates.
New method uses kernel Stein discrepancy for measure transport without strict continuity constraints.
problem Minimizing Kullback-Leibler divergence for posterior approximation.
method Proposes minimizing kernel Stein discrepancy instead of Kullback-Leibler divergence.
result Demonstrates consistency and competitiveness of the new method.
Paper proposes kernelized Stein tests for time-to-event data with censoring.
problem Testing goodness-of-fit for time-to-event data with censoring.
method Combining Stein's method and kernelized discrepancies for non-parametric testing.
result Proposed kernelized Stein discrepancy tests perform better than existing methods.
ConvMMD improves inference in noisy data.
problem Inference degradation due to measurement error in noisy data.
method Convolutional Maximum Mean Discrepancy (convMMD) for inference with noisy, heteroscedastic observations.
result Established consistency and asymptotic normality of the convMMD-based estimator.
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.
Improved kernel Stein discrepancy for large-scale data.
problem Efficiently testing probability distributions with kernel methods.
method Nyström approximation to reduce runtime complexity.
result Nyström-based KSD is n \sqrt{n} n -consistent and applicable for large datasets. Study on discrepancy principle for learning algorithms in nonparametric regression.
problem Determining optimal iteration number in nonparametric regression with unknown optimal iteration.
method Investigates discrepancy principle and modified principles for kernelized spectral filters, using deviation inequalities and change-of-norm arguments.
result Classical discrepancy principle is adaptive for slow rates, while modified principles are adaptive for faster rates.
New kernel-based method detects biased MCMC samples.
problem Detecting biased MCMC samples with standard diagnostics.
method Kernel Stein discrepancy (KSD) based on Stein's method.
result KSDs can detect convergence for various target distributions.
A new test compares latent variable models with kernel methods.
problem Comparing latent variable models with intractable marginal distributions.
method Kernel Stein Test, calibrated threshold, low-dimensional latent structure exploitation.
result Significantly outperforms Maximum Mean Discrepancy test in certain latent structure cases.
We consider the problem of improving the efficiency of randomized Fourier feature maps to accelerate training and testing speed of kernel methods on large datasets. These approximate feature maps arise as Monte Carlo approximations to integral representations of shift-invariant kernel functions (e.g., Gaussian 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.
The paper improves Stein importance sampling for Markov chain samples.
problem Improving the accuracy of sampling from complex distributions.
method Reproducing Stein kernels approach for post-hoc correction.
result Consistent estimators for target distributions using geometrically ergodic Markov chains.
Improved convergence rates for Stein Variational Gradient Descent in finite-particle settings.
problem Improving convergence rates for Stein Variational Gradient Descent in finite-particle settings.
method Analyzing the time derivative of relative entropy and splitting it into dominant and smaller parts.
result Finite-particle convergence rates of order 1/\sqrt{N} for Kernelized Stein Discrepancy and Wasserstein-2 metrics.
A new method calculates intrinsic effective sample size for manifold-valued data.
problem Challenges in choosing effective sample size for manifold-valued data.
method Proposes an intrinsic effective sample size based on kernel discrepancy.
result Establishes an exact finite-sample risk interpretation and consistency of the estimator.
Unified score and distance-based GoF tests for model adequacy.
problem Difficulty in extending score-based GoF tests to nonparametric alternatives.
method Introducing semiparametric kernelized Stein discrepancy (SKSD) test.
result SKSD test is computationally efficient and universally consistent.
New method uses multiple kernels to improve SVGD performance.
problem Sub-optimal performance of single kernel in SVGD.
method Combines multiple kernels to approximate optimal kernel, using Kernelized Stein Discrepancy (KSD) and constructing Multiple Kernel SVGD (MK-SVGD).
result Consistently matches or outperforms competing methods in experiments.
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.
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.