Paper generalizes regression problems in hyper-RKHS for kernel learning.
problem Kernel learning and out-of-sample extensions in regression problems.
method Introduces two regularized regression models in hyper-RKHS, including KRR and SVR, and applies divide-and-conquer with Nyström approximation for scalability.
result Proves asymptotic convergence results and derives learning rates for regularized regression algorithms in hyper-RKHS.
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on Lp spaces and sets of measures. Kernel methods are studied in a mean field limit for high-dimensional data.
problem Analyzing kernel methods in high-dimensional data with many variables.
method Investigation of kernel methods in the mean field limit of interacting particle systems.
result Rigorous mean field limit of kernels and detailed analysis of the limiting reproducing kernel Hilbert space.
A new matrix Hilbert space framework preserves matrix structure and captures multi-way correlations.
problem Preserving matrix structure and capturing multi-way correlations in matrix learning.
method Introducing matrix Hilbert space and reproducing kernel matrix Hilbert space (RKMHS) to perform matrix inner product space.
result Preserves matrix structure and captures multi-way correlations without tensor decomposition.
Study RKHS on manifolds, linking Sobolev and diffusion spaces.
problem Characterizing RKHS on manifolds and their properties.
method Analyzing Sobolev and diffusion spaces on Riemannian manifolds.
result Sobolev spaces are RKHS under certain conditions, and diffusion spaces are introduced.
Quantum models use complex Hilbert spaces for uncertainty.
problem Modeling dynamics in continuous-valued features.
method Quantum Graphical Models (QGMs) and Hilbert Space Embedding (HSE).
result HSE-HQMMs are competitive with state-of-the-art models.
Study on how sampling works for complex data functions.
problem Analyzing convergence of sampling algorithms for RKHS functions.
method Minimalistic assumptions on kernel and data, error estimates in RKHS norm, uniform convergence on compact domains.
result New convergence rates for Lipschitz and Hölder continuous kernels.
Optimizes learning Hilbert-Schmidt operators between Sobolev spaces.
problem Statistical limits of learning mappings between infinite-dimensional function spaces.
method Minimax optimal regularization and multilevel training.
result Multilevel kernel operator learning achieves optimal learning rate.
New recursive algorithm estimates conditional kernel mean embeddings in Hilbert space.
problem Estimating conditional distributions in RKHS for supervised learning.
method Recursive algorithm in L2 space for conditional kernel mean map. result Strong L2 consistency of recursive estimator proved. Embeds persistence diagrams into Hilbert spaces to use kernel methods.
problem No inner product structure on persistence diagrams.
method Shows non-embeddability of persistence diagrams into Hilbert spaces.
result Persistence diagrams with bottleneck distance do not coarse embed into Hilbert spaces.
Paper develops a dual formulation for PCA in Hilbert spaces.
problem Characterizing probabilistic PCA in Hilbert spaces.
method Dual formulation for probabilistic PCA in Hilbert spaces.
result Generative framework for kernel methods developed.
Kernelized cumulants improve statistical analysis in high-dimensional spaces.
problem Statistical analysis in high-dimensional spaces with low variance estimators.
method Extending cumulants to RKHS using tensor algebra and kernel trick.
result Kernelized cumulants provide new all-purpose statistics with computational tractability.
A new method estimates multi-dimensional value distributions using Hilbert space embeddings.
problem Estimating value distributions in complex, multi-dimensional reinforcement learning settings.
method Hilbert space mappings and kernel mean embeddings to estimate the kernel mean embedding of multi-dimensional value distributions.
result Uniform convergence guarantees and robust off-policy evaluation demonstrated in simulations.
The paper develops a uniform function estimator in RKHS for regression.
problem Reconstructing functions from noisy data at random locations.
method Using reproducing kernel Hilbert spaces and Gaussian random fields.
result The estimator converges uniformly to the conditional expectation.
Survey of kernels, RKHS, and their applications in machine learning.
problem Understanding kernels and their applications in machine learning.
method Review of historical context, mathematical definitions, and practical applications of kernels.
result Comprehensive overview of kernels, RKHS, and their applications.
The paper develops methods to handle missing data using regularized M-estimation in reproducing kernel Hilbert space.
problem Handling missing data in statistical analysis.
method Kernel ridge regression for imputation and maximum entropy method for propensity score estimation.
result The proposed methods achieve statistical consistency and asymptotic equivalence.
Kernel Autoencoder (KAE) encodes any data type using RKHSs.
problem Representing any data type in a compact form.
method KAE uses mappings from vv-RKHSs to minimize reconstruction error.
result KAE can autoencode any kind of data by choosing X as a RKHS.
This note explains when neural networks can be seen as Gaussian processes.
problem Understanding the relationship between neural networks and Gaussian processes.
method Formulating a Gaussian process regression based on neural network outputs and analyzing the resulting posterior mean functions.
result The posterior mean functions of neural networks follow a Gaussian process in certain cases, providing an interpretation of reproducing kernel Hilbert spaces.
Online machine learns from signals robustly using consensus optimization.
problem Signal classification in online settings.
method Reproducing kernel Hilbert space with consensus optimization.
result Robust signal classification achieved.
New theoretical tools simplify kernel-based tests analysis.
problem Asymptotic behavior of kernel-based tests in various scenarios.
method Avoids complex expansions and limit theorems, works directly with Hilbert spaces random functionals.
result Framework leads to simpler analysis with minimal regularity conditions.
Study entropic regularization of Gaussian measures and processes on Hilbert space.
problem Regularizing 2-Wasserstein distance for infinite-dimensional Gaussian measures and processes.
method Minimum Mutual Information property, closed form formulas, Fréchet differentiability, Sinkhorn barycenter equation.
result Entropic 2-Wasserstein distance and Sinkhorn divergence are Fréchet differentiable in Hilbert space.
The study characterizes kernel spaces on hyperspheres, impacting cubature algorithms.
problem Characterizing kernel spaces on hyperspheres for cubature algorithms.
method Characterization of Sobolev spaces and reproducing kernel Hilbert spaces over hyperspheres.
result Direct consequences for kernel cubature and worst-case error rates.
The article analyzes LCE in Hilbert space, deriving new formulas and regularisation methods.
problem Analyzing conditional expectation in infinite-dimensional Hilbert space.
method Establishing analytical properties and regularisation for LCE in Hilbert space, deriving new formulas.
result Simple derivation and intuitive justification of conditional mean embedding formula.
A fast method for selecting Gaussian kernel bandwidth in kernel-based classifiers.
problem High computational complexity in estimating Gaussian kernel bandwidth.
method Developed based on reproducing kernel Hilbert space operators.
result Proposed method outperforms state-of-the-art methods in computational time and performance.
The paper proposes a Gaussian mixture model for Hilbert-space-valued data.
problem Challenges in characterizing probability measures for infinite-dimensional random objects.
method Gaussian mixture framework based on kernel mean embeddings.
result The proposed algorithm yields a dense class of approximations in infinite-dimensional spaces.
Paper introduces RKHM for more explicit variable structures analysis.
problem Explicitly analyzing structures among variables.
method Orthonormal systems in Hilbert C∗-modules, RKHM. result Theoretical and practical procedures for RKHM orthonormalization.
Kernel interpolation is inconsistent for norms with smoothness above a constant.
problem Inconsistency of kernel interpolation in reproducing kernel Hilbert spaces.
method Lower bounds for generalization error in Sobolev norms.
result Kernel interpolation is always inconsistent for norms with smoothness above a constant.
Estimates kernel eigenvalues for compositional dot-product kernels.
problem Improving estimates for kernel eigenvalues.
method Eigenvalue decay estimates of integral operators associated with dot-product kernels.
result Improved estimates for kernel volumes in reproducing kernel Hilbert spaces.
This paper introduces Bayes Hilbert spaces for efficient posterior approximation.
problem Efficient posterior approximation in Bayesian models for large datasets.
method Develops Bayes Hilbert spaces for posterior approximation and connects them to Bayesian coresets and kernel-based distances.
result Bayes Hilbert spaces provide a novel framework for posterior approximation that is computationally efficient.
Kernel VICReg improves SSL in RKHS, capturing nonlinear structures.
problem Limited ability of existing SSL methods to handle nonlinear dependencies.
method Kernel VICReg framework in RKHS, kernelizing VICReg objectives.
result Kernel VICReg mitigates representational collapse and improves performance.
Stochastic Gradient Descent improved for various Hilbert scales and misspecified models.
problem Understanding and optimizing SGD in Hilbert scales for machine learning.
method Extending SGD analysis to Hilbert scales, including Sobolev and Diffusion spaces, and showing the effects of smoothness and preconditioning.
result Violation of smoothness assumption affects learning rate; preconditioning in Hilbert scales reduces the number of iterations for misspecified models.
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.
Lecture notes on kernel functions and Random Fourier Features.
problem Understanding and approximating kernel functions in machine learning.
method Mathematical background and proofs of concentration results.
result Estimation of error in Random Fourier Features approximation.
New algorithm improves dynamic mode decomposition for high-dimensional data.
problem Reduced modeling in high-dimensional spaces.
method Low rank constraint optimization and kernel-based computation.
result Gain in approximation accuracy and computational efficiency.
Subsampling reduces computational cost in supervised learning in reproducing kernel Hilbert spaces.
problem Reducing computational cost in supervised learning
method Subsampling minimizes empirical risk in reproducing kernel Hilbert spaces
result Optimal subsampling scheme revealed
The paper shows how multi-task learning in neural networks is similar to kernel regression and Hilbert spaces.
problem Understanding the solutions to multi-task shallow ReLU neural network learning problems.
method Analyzing the properties of solutions to multi-task shallow ReLU neural network learning problems, proving uniqueness and equivalence to minimum-norm interpolation problems in Hilbert spaces.
result The solutions to multi-task neural network interpolation problems are almost always unique and coincide with the solution to a minimum-norm interpolation problem in a Sobolev (Reproducing Kernel) Hilbert Space.
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. New algorithms learn in complex decision-making problems with smooth transitions.
problem Learning in complex decision-making problems with smooth transitions.
method UCB and PSRL philosophies applied to episodic Markov decision processes with kernel approximation.
result Low regret learning achieved in continuous state and action spaces.
Efficient estimators for smooth Hilbert-valued parameters with theoretical guarantees.
problem Estimating smooth Hilbert-valued parameters with theoretical guarantees.
method Pathwise differentiable Hilbert-valued parameters, efficient influence functions, regularized one-step estimators.
result Theoretical guarantees for efficient estimators even when nuisance functions are arbitrary.
Solves low-rank approximation problems in Hilbert spaces.
problem Low-rank approximation in Hilbert spaces.
method Closed-form solutions and error bounds for bounded linear operators.
result Generalization to bounded linear operators from finite dimensions.
This work explores the connection between distances and kernels for conditional independence.
problem Measuring conditional independence in various fields like causal discovery and feature selection.
method Investigates the relationship between conditional independence measures induced by distances and reproducing kernels.
result Some kernel-based conditional independence measures are not equivalent to distance-based measures.
Paper introduces RKHM and KME for richer data analysis.
problem Lack of rich data structures in kernel methods.
method Proposes RKHM and KME for functional data analysis.
result RKHM captures structural properties in functional data.
Kernel-based Bayesian filter for nonlinear systems using infinite-dimensional operators.
problem Modeling and predicting nonlinear dynamical systems.
method Functional Bayesian perspective, reproducing kernel Hilbert space, Gaussian kernel.
result Effective approximation and accurate results for nonlinear systems.
KT combines treelets with kernel functions for hierarchical clustering.
problem Hierarchical clustering of non-numeric data.
method Combines treelets and kernel functions to handle non-numeric data.
result KT effectively clusters non-numeric data.
New algorithm optimizes functions in Matérn kernel RKHS with noisy feedback.
problem Optimizing functions in RKHS of Matérn kernel with noisy bandit feedback.
method π-GP-UCB algorithm with guaranteed sublinear regret for all ν > 1 and d ≥ 1.
result First practical approach with guaranteed sublinear regret for all ν > 1 and d ≥ 1.
This study connects Gaussian processes and RKHS, bridging two machine learning communities.
problem Understanding the relationship between Gaussian processes and RKHS.
method Examining connections and equivalences in regression, interpolation, and other topics.
result Established the equivalence between Gaussian Hilbert space and RKHS.
A theorem for Hilbert space-valued functions simplifies machine learning models.
problem Learning functions in Hilbert space-valued spaces.
method Generalized Representer Theorem for Hilbert space-valued functions.
result Unified view of supervised and semi-supervised learning methods.