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.
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.
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.
The maximum mean discrepancy (MMD) is a kernel-based distance between probability distributions useful in many applications (Gretton et al. 2012), bearing a simple estimator with pleasing computational and statistical properties. Being able to efficiently estimate the variance of this estimator is very helpful to vario…
High-dimensional U-statistics show surprising phase transitions, impacting kernel-based tests.
problem Understanding phase transitions in high-dimensional U-statistics.
method Proved a convergence theorem for U-statistics of degree two in high dimensions.
result High-dimensional U-statistics can have non-Gaussian limits with larger variance and asymmetry.
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…
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.
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.
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.
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.
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…
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.
A new kernel-based nonconformity score improves multivariate prediction regions.
problem Tackling the challenge of compressing multivariate residual vectors into scalars while preserving geometric structure.
method Introducing a Multivariate Kernel Score (MKS) that decomposes into an anisotropic MMD, providing finite-sample coverage guarantees and convergence rates.
result The MKS produces prediction regions that explicitly adapt to geometric structure, reducing volume compared to ellipsoidal baselines.
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.
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.
Learning with kernels is an important concept in machine learning. Standard approaches for kernel methods often use predefined kernels that require careful selection of hyperparameters. To mitigate this burden, we propose in this paper a framework to construct and learn a data-dependent kernel based on random features …
The paper introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely-many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorpora…
Quantum probability metrics improve distribution comparison in high dimensions.
problem Challenges in comparing probability distributions, especially in high-dimensional and non-compact domains.
method Quantum probability metrics (QPMs) derived from quantum state spaces, overcoming limitations of MMD.
result QPMs offer enhanced sensitivity to subtle distributional differences in high dimensions and improve performance in generative modeling.
The paper introduces new KMEs to capture stochastic process filtrations.
problem Missing filtration information in stochastic processes.
method Higher order kernel mean embeddings (KMEs) conditioned on filtrations.
result Consistent estimators and tests for filtration-sensitive information.
The paper improves probabilistic herding methods using Gibbs distributions.
problem Improving integration accuracy over Monte Carlo quadrature in infinite-dimensional RKHS.
method Developed a Gibbs distribution over quadrature nodes to minimize MMD.
result The Gibbs distribution outperforms i.i.d. Monte Carlo in integration accuracy.
KCUSUM detects abrupt changes in real-time data streams efficiently.
problem Detecting abrupt changes in high-volume scientific data streams.
method Kernel-based Cumulative Sum (KCUSUM) algorithm using Maximum Mean Discrepancy (MMD).
result KCUSUM outperforms traditional CUSUM in online change point detection.
Improved image generation with private data using perceptual features.
problem Difficulty in training generative models with differential privacy.
method Use pre-trained perceptual features to learn private data distribution.
result Generative models can generate high-quality images with low privacy budget (ϵ≈2). A novel kernel-based test detects equality versus singularity of two probability measures.
problem Detecting equality versus singularity of two probability distributions.
method Combines kernel mean and kernel covariance embeddings to construct a likelihood ratio test statistic.
result The test statistic satisfies a '0/\infty' law, vanishing under the null and diverging under the alternative.
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.
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.
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.
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.
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.
Dependent MMD coresets help compare multiple related datasets.
problem Comparing multiple related datasets for insights into model generalization.
method Dependent MMD coresets for collections of datasets.
result Dependent MMD coresets facilitate comparison and understanding of multiple related datasets.
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.
Maximum mean discrepancy (MMD) has been widely adopted in domain adaptation to measure the discrepancy between the source and target domain distributions. Many existing domain adaptation approaches are based on the joint MMD, which is computed as the (weighted) sum of the marginal distribution discrepancy and the condi…
MMD-Flagger detects hallucinations in LLMs by tracking MMD between outputs and temperature-generated counterparts.
problem Detecting hallucinations in large language models.
method Maximum Mean Discrepancy (MMD) to track the difference between model outputs and temperature-generated counterparts.
result MMD-Flagger detects most hallucinations by analyzing the shape of the MMD trajectory.
kdiff measures distances for time series and structured data.
problem Estimating distances between time series and structured data.
method kdiff uses non-linear kernel distances based on matching overlapping distributions.
result kdiff is more robust to noise and partial occlusions.
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…
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.
A new, fast kernel test for large data.
problem Efficient kernel two-sample tests for high-dimensional, large-scale data.
method A new kernel-based test that is computationally efficient and robust to high dimensions.
result The new test performs well across various alternatives and dimensions.
Paper optimizes PCA for fairness using MMD and Stiefel manifold optimization.
problem Fair principal component analysis (PCA) to minimize MMD between protected classes.
method Formulates fair PCA as non-convex optimization over Stiefel manifold, solves using REPMS with theoretical guarantees.
result Our approach outperforms prior work in fairness, explained variance, and runtime.
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.
Framework identifies causal direction from single data setting.
problem Identify causal direction from single observational data.
method VCEI framework based on ICM principle and artificial variation.
result VCEI is competitive to other frameworks in identifying causal direction.
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.
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.
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.
Generative adversarial nets (GANs) are widely used to learn the data sampling process and their performance may heavily depend on the loss functions, given a limited computational budget. This study revisits MMD-GAN that uses the maximum mean discrepancy (MMD) as the loss function for GAN and makes two contributions. F…
Improved MMD test for two-sample testing with random Fourier features.
problem Quadratic-time complexity of MMD test for large-scale analysis.
method Approximated MMD test using random Fourier features, investigating time-power trade-off.
result Sub-quadratic time complexity with same minimax separation rates as MMD test.
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.
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 …
New method improves MMD estimation without convexity assumptions.
problem Lack of theoretical guarantees for MMD estimation algorithms.
method Preconditioned gradient descent (PGD) scheme for MMD optimization.
result PGD scheme converges globally under specific conditions.
New framework improves model reliability under distribution shifts.
problem Lack of formal guarantees connecting shift magnitude to prediction reliability in TTA methods.
method Develops a PAC-Bayesian framework interpreting MMD-balls as credal sets.
result Establishes generalization bounds and provides epistemic uncertainty quantification.