Novel RKHS approach solves complex financial model equations.
problem Calibrating singular local stochastic volatility models.
method Reproducing Kernel Hilbert Space (RKHS) regularization.
result Regularized model is well-posed and replicates option prices.
Enhanced kernel ridgeless regression improves performance with LAB RBF kernels.
problem Lack of flexibility in kernel ridgeless regression.
method Locally-Adaptive-Bandwidths (LAB) RBF kernels and kernel learning techniques.
result Functions learned from LAB RBF kernels belong to an integral space of RKHSs, demonstrating robust generalization.
New tests for distributional causal effects using improved kernel estimators.
problem Testing for higher-order moments and multidimensional outcomes affected by treatment.
method Improved kernel estimators based on doubly robust mean embeddings.
result New permutation-based tests for distributional causal effects with improved convergence rates.
New algorithm optimizes noisy, potentially corrupted functions.
problem Optimizing unknown functions with noisy bandit feedback, especially when evaluations are corrupted.
method Fast-Slow GP-UCB algorithm, combining robust and non-robust evaluations, enlarged confidence bounds.
result Theoretical analysis upper bounds cumulative regret, showing dependencies on corruption level and kernel.
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.
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. 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.
Theoretical studies have proven that the Hilbert space has remarkable performance in many fields of applications. Frames in tensor product of Hilbert spaces were introduced to generalize the inner product to high-order tensors. However, these techniques require tensor decomposition which could lead to the loss of infor…
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.
We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with…
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.
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.
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.
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.
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 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.
Gradient descent in neural networks analyzed using RKBS for broader applicability.
problem Analyzing neural network training in the over-parametrized limit.
method Constructing an exact power-series representation of neural networks in RKBS, proving replicability of gradient descent sequences.
result Gradient descent sequences can be exactly replicated by regularized sequential learning in RKBS, providing new theoretical insights.
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.
Quantum Graphical Models (QGMs) generalize classical graphical models by adopting the formalism for reasoning about uncertainty from quantum mechanics. Unlike classical graphical models, QGMs represent uncertainty with density matrices in complex Hilbert spaces. Hilbert space embeddings (HSEs) also generalize Bayesian …
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.
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. 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.
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.
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.
Study Markov chain gradient descent in Hilbert spaces for quadratic loss.
problem Approximating optimal solutions for quadratic loss functions.
method Developed a Markov chain-based stochastic gradient algorithm in Hilbert spaces.
result Established probabilistic upper bounds on convergence.
This paper investigates a novel algorithmic approach to data representation based on kernel methods. Assuming that the observations lie in a Hilbert space X, the introduced Kernel Autoencoder (KAE) is the composition of mappings from vector-valued Reproducing Kernel Hilbert Spaces (vv-RKHSs) that minimizes the expected…
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.
Researchers approximate conditional expectation operators using kernel methods.
problem Statistical approximation of conditional expectation operators under minimal assumptions.
method Modifying the domain of the operator, approximating it by Hilbert-Schmidt operators in a reproducing kernel Hilbert space.
result The nonparametric estimate of the operator converges to a specific limiting object.
Improved estimation of higher order integrals using shrinkage techniques.
problem Estimating higher order Bochner integrals in non-parametric settings.
method Shrinkage of U-statistic towards a target element, considering kernel degeneracy.
result Consistent shrinkage estimators with fast rates of convergence, even for non-degenerate kernels.
The paper develops divergences for Gaussian processes and RKHS settings.
problem Estimating divergences in infinite-dimensional spaces.
method Formulations of Alpha Log-Det divergences, continuity in norm, laws of large numbers, consistent estimation from finite samples.
result Infinite-dimensional divergences can be estimated from finite-dimensional versions with dimension-independent sample complexities.
We study the stability properties of nonlinear multi-task regression in reproducing Hilbert spaces with operator-valued kernels. Such kernels, a.k.a. multi-task kernels, are appropriate for learning prob- lems with nonscalar outputs like multi-task learning and structured out- put prediction. We show that multi-task ke…