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.
Paper proves convergence rates for Gaussian kernel ridge regression.
problem Understanding convergence rates for Gaussian kernel ridge regression.
method Establishes polynomial convergence rates for KRR with fixed hyperparameters.
result First polynomial convergence rates for Gaussian kernel ridge regression.
CKA with Gaussian RBF kernels converges linearly as bandwidth increases.
problem Understanding the behavior of CKA with large bandwidth Gaussian kernels.
method Analyzing the convergence of CKA based on Gaussian RBF kernels in the large-bandwidth limit.
result CKA based on Gaussian RBF kernels converges linearly as bandwidth increases.
Study guarantees convergence of mean shift mode estimation.
problem Ensuring reliable mode estimation in KDE using mean shift.
method Utilizes Łojasiewicz inequality to prove convergence rate.
result Extends convergence guarantees to biweight kernel.
Paper studies t-SNE convergence with generalized kernels.
problem Understanding convergence of t-SNE with generalized kernels.
method Concrete formulation of generalized kernels, proving convergence to an equilibrium distribution.
result t-SNE converges to an equilibrium distribution under certain conditions for generalized kernels.
The paper extends kernel ridge regression to product kernels and reveals new convergence behaviors.
problem Understanding kernel ridge regression in large dimensions with various kernels.
method Established a broad family of large dimensional kernels and derived convergence rates.
result Revealed new phenomena including minimax optimality, saturation effect, and multiple descent behavior.
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.
Paper proves MS convergence for radially symmetric kernels with large bandwidths.
problem Proving convergence of mean shift algorithm with radially symmetric kernels.
method Analyzes convergence of mean shift algorithm with radially symmetric, positive definite kernels.
result Guaranteed convergence for sufficiently large bandwidth in any dimension.
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.
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 simulation method for Volterra processes improves convergence for rough kernels.
problem Simulating Volterra processes with singular kernels.
method iVi (integrated Volterra implicit) scheme based on Inverse Gaussian distribution.
result The iVi scheme achieves weak convergence with few time steps, especially for rough kernels.
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.
Ad-SVGD optimizes kernel parameters for SVGD, improving inference performance.
problem Efficiently approximating posterior distributions in Bayesian inference.
method Adaptive kernel selection for SVGD dynamics.
result Ad-SVGD outperforms standard heuristics in various tasks.
Kernel learning FBSDE filter improves nonlinear filtering efficiency.
problem Nonlinear filtering problem in high-dimensional systems.
method Iterative and adaptive meshfree approach using forward backward SDE and KDE.
result Rigorous convergence analysis provided, supporting empirical results.
Study continuity of Bergman kernels on degenerating varieties.
problem Continuity and uniform convergence of Bergman kernels on degenerating varieties.
method Introduced fiberwise Bergman kernel for flat families of polarized varieties, established continuity and uniform convergence results.
result Uniform convergence of Fubini-Study currents and continuity of fiberwise Bergman kernel on test configurations.
Averaged SGD achieves optimal convergence rate for neural networks in the NTK regime.
problem Convergence analysis of averaged stochastic gradient descent for neural networks.
method Analyzed convergence of averaged stochastic gradient descent for overparameterized two-layer neural networks.
result Achieved minimax optimal convergence rate with global convergence guarantee.
Improved kernel herding algorithm for faster quadrature rule convergence.
problem Slow convergence speed of standard kernel herding algorithm.
method Improved gradient approximation to obtain sparser solutions.
result The cosine of the angle between negative gradient and approximate gradient determines convergence speed.
The paper proves convergence of graph Laplacian with kNN self-tuned kernels.
problem Theoretical and practical challenges in choosing kernel bandwidth for graph-based analysis.
method Develops and analyzes a new family of kNN self-tuned kernels for graph Laplacian convergence.
result Proves convergence of graph Laplacian to manifold Laplacian for new kNN self-tuned kernels.
Study shows deterministic equivalent for neural network kernel convergence.
problem Understanding convergence of neural network kernels.
method Analyzes empirical spectral distribution of Conjugate Kernel, proving convergence to a deterministic limit.
result Obtains a deterministic equivalent for the Stieltjes transform and resolvent of the Conjugate Kernel.
The paper examines the neural tangent kernel for PINNs solving general PDEs and finds convergence conditions.
problem Analyzing the convergence of neural tangent kernel for PINNs solving general PDEs.
method Analysis of NTK initialization and convergence during training for general PDEs using PINNs.
result Homogeneity of differential operators is crucial for NTK convergence.
We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensio…
Faster convergence of kernel mean embeddings using variance information.
problem Speeding up the convergence rate of kernel mean embeddings.
method Leveraging variance information in reproducing kernel Hilbert space and estimating variance from data.
result Efficiently estimate variance information from data to achieve distribution-agnostic convergence bounds.
Study on Neural Tangent Kernel of Matrix Product States and their convergence.
problem Understanding the convergence of Neural Tangent Kernel of Matrix Product States.
method Analyzing the Neural Tangent Kernel of Matrix Product States and proving its convergence in the infinite bond dimensional limit.
result The Neural Tangent Kernel of Matrix Product States converges to a constant matrix during training.
Kernel method improves instrumental variable regression rates.
problem Nonparametric instrumental variable regression with weak instruments.
method Kernel-based two-stage least-squares method, strong L2 convergence analysis. result Minimax optimal rates for instrumental regression under standard assumptions.
Study on SGD for overparameterized neural networks, focusing on convergence rates.
problem Understanding convergence rates of SGD in overparameterized two-layer neural networks.
method Combines NTK approximation with RKHS analysis to explore SGD dynamics.
result Established sharp convergence rates for SGD in overparameterized two-layer neural networks.
Active data collection improves convergence rates in operator learning.
problem Improving convergence rates in operator learning with linear target and stochastic input.
method Active data collection strategies with mean-zero stochastic process and continuous covariance kernels.
result Achieves arbitrarily fast error convergence rates with eigenvalue decay of covariance kernels.
New KSDs control moments in approximations, improving diagnostics and tests.
problem Inability of standard KSDs to control moment convergence.
method Developed alternative diffusion KSDs under sufficient conditions.
result First KSDs to exactly characterize q-Wasserstein convergence.
The paper studies convergence of kernel autocovariance operators for stationary processes.
problem Estimating autocovariance operators of stationary processes on Polish spaces.
method Investigates convergence of empirical estimates of autocovariance operators under various conditions.
result Provides consistency results for kernel PCA and spectral analysis methods.
Study shows convergence of Bergman kernels on covering spaces of Kähler manifolds.
problem Analyzing convergence of Bergman kernels on covering spaces of Kähler manifolds.
method Proving convergence of Bergman kernels and L2-Hodge numbers on a tower of coverings. result Sections of canonical line bundles give rise to immersions into projective spaces.
Paper analyzes convergence rates of mean-field SVGD method.
problem Establishing quantitative rates of convergence for mean-field SVGD.
method Quantitative analysis of mean-field SVGD dynamics on torus.
result Explicit polynomial convergence rates in L2-norm for Riesz-type kernels.
This work closes the theory-practice gap for distributed optimization methods by introducing a new regularity condition.
problem Existing convergence conditions for distributed optimization methods are violated by nearly all kernels used in practice.
method Introduces Hessian relative uniform continuity (HRUC) to guarantee convergence under mild conditions.
result Derives convergence guarantees for mirror descent-based gradient tracking without restrictive assumptions.
New method guarantees global convergence in variational inference.
problem Limited convergence to local optima in variational inference.
method Minimizes inclusive KL divergence using neural networks and neural tangent kernel.
result Gradient descent dynamics converge to a unique solution in function space.
Two-layer neural networks learn efficiently using kernel methods in mean-field analysis.
problem Feature learning ability of two-layer neural networks in the mean-field regime.
method Mean-field analysis through kernel methods, focusing on dynamics of the first layer's kernel.
result Two-layer neural networks can learn a union of multiple reproducing kernel Hilbert spaces more efficiently than kernel methods.
Approximate Markov chain Monte Carlo (MCMC) offers the promise of more rapid sampling at the cost of more biased inference. Since standard MCMC diagnostics fail to detect these biases, researchers have developed computable Stein discrepancy measures that provably determine the convergence of a sample to its target dist…
The rate of convergence of weighted kernel herding (WKH) and sequential Bayesian quadrature (SBQ), two kernel-based sampling algorithms for estimating integrals with respect to some target probability measure, is investigated. Under verifiable conditions on the chosen kernel and target measure, we establish a near-geom…
This paper extends mirror descent to Banach spaces with reproducing kernels.
problem Optimizing in Banach spaces with reproducing kernels.
method Mirror descent algorithm adapted for Banach spaces with reproducing kernels.
result Mirror descent achieves linear convergence in certain conditions and standard convergence in a constrained setting.
Wide residual networks generalize well with uniform convergence to RNTK as width increases.
problem Understanding the generalization ability of wide residual networks.
method Uniform convergence of residual network kernel to residual neural tangent kernel (RNTK).
result Generalization error converges to kernel regression error with respect to RNTK.
We theoretically investigate the convergence rate and support consistency (i.e., correctly identifying the subset of non-zero coefficients in the large sample limit) of multiple kernel learning (MKL). We focus on MKL with block-l1 regularization (inducing sparse kernel combination), block-l2 regularization (inducing un…
The paper proves spectral convergence rates for graph Laplacian to manifold Laplace-Beltrami operator.
problem Spectral convergence of graph Laplacian to manifold Laplace-Beltrami operator.
method Analysis of Dirichlet form convergence and construction of approximate eigenfunctions via manifold heat kernel.
result Proves spectral convergence rates for Gaussian kernelized graph Laplacian.
Paper improves heat kernel estimates on Ricci shrinkers.
problem Estimates on heat kernels for Ricci shrinkers.
method Improves estimates from previous work and extends recent progress.
result Theory of $\IF$-convergence holds on Ricci flows induced by Ricci shrinkers.
Study local convergence of GDA for training GANs with kernel-based discriminators.
problem Analyzing the local dynamics of GDA for GANs with kernel-based discriminators.
method Linearization of a non-linear dynamical system, under an isolated points model assumption.
result Showed phase transitions indicating convergence, oscillation, or divergence of GDA.
In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of RCD∗(K,N)-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in RCD∗(K,N)-spaces. We use then these results to initiate the study of Weyl's law in the RCD setting
Study shows how heavy-tailed Hawkes processes can model rough volatility in financial markets.
problem Modeling rough volatility in financial markets with heavy-tailed Hawkes processes.
method Established weak convergence of Hawkes process with power-law kernel, derived scaling limit for financial market model.
result Price-volatility process converges weakly to a rough Heston model after rescaling.
Spectral algorithms on manifolds using diffusion kernels improve convergence rates.
problem The limitations of existing spectral algorithms in RKHSs for data on manifolds.
method Integrating manifold structure into spectral algorithms using heat kernel diffusion spaces.
result Spectral algorithms converge to the target function and its derivatives in a strong sense, with rates dependent on manifold intrinsic dimension.
Study birth-death dynamics for sampling Gibbs measures with nonconvex potentials.
problem Sampling Gibbs measures with nonconvex potentials.
method Birth-death dynamics, Kullback-Leibler divergence, χ2 divergence, kernel-based approximations, Γ-convergence of gradient flows. result Probability density converges exponentially fast to Gibbs equilibrium measure with a universal rate.
In this paper, we study the problem of sparse multiple kernel learning (MKL), where the goal is to efficiently learn a combination of a fixed small number of kernels from a large pool that could lead to a kernel classifier with a small prediction error. We develop an efficient algorithm based on the greedy coordinate d…
The paper explores identifiability and stability in drifting fields using companion-elliptic kernels.
problem Identifying and stabilizing drifting fields in generative modeling.
method Introduces companion-elliptic kernel families and analyzes their properties to address identifiability and stability issues.
result Established field identifiability for arbitrary Borel probability measures and demonstrated that field convergence alone does not guarantee weak convergence.
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.