Paper calculates eigenvalue decay rates for neural network kernels on general domains.
problem Determining eigenvalue decay rates for neural network kernels on arbitrary domains.
method Proved dynamics of wide neural networks approximates NTK on general domains, used minimax optimality and interpolation spaces.
result Provided strategy to calculate eigenvalue decay rates for neural network kernels.
We compute the leading and sub-leading terms in the asymptotic expansion of the Szegö kernel on the diagonal of a class of pseudoconvex Reinhardt domains whose boundaries are endowed with a general class of smooth measures. We do so by relating it to a Bergman kernel over projective space.
Framework for domain adaptation using pseudo-labels from unlabeled data.
problem Improving prediction accuracy in target domain with covariate shift.
method Kernel GLMs with labeled and pseudo-labeled data, using imputation model for target data.
result Non-asymptotic excess-risk bounds for effective labeled sample size.
Needlets have been recognized as state-of-the-art tools to tackle spherical data, due to their excellent localization properties in both spacial and frequency domains. This paper considers developing kernel methods associated with the needlet kernel for nonparametric regression problems whose predictor variables are de…
D2V learns domain-specific embeddings for domain generalization.
problem Learning decision functions across multiple related domains with limited labeled data.
method Proposes a neural network architecture, Domain2Vec (D2V), that learns domain-specific embeddings and uses them for generalization.
result D2V outperforms other algorithms in domain generalization tasks for image classification.
The paper tackles domain generalization using functional regression.
problem Learning a model that generalizes well across different source distributions.
method Functional regression approach to learn a linear operator between marginal and conditional distributions.
result The proposed algorithm achieves finite sample error bounds for the idealized risk.
Efficiently scales continuous kernels with sparse Fourier domain learning.
problem High computational and memory demands, spectral bias in continuous kernels.
method Sparse learning in the Fourier domain.
result Efficient scaling of continuous kernels, reduced computational and memory requirements, mitigated spectral bias.
The paper studies invariant weighted Bergman metrics on domains.
problem Investigating invariant weighted Bergman metrics under biholomorphisms.
method Introducing invariant weight assignments, using Bergman's minimum integral method and domain version of Tian-Yau-Zelditch expansion.
result Uniform convergence of weighted Bergman kernels and metrics on uniform squeezing domains.
The study shows algebraic Bergman kernels imply finite type boundaries in complex domains.
problem Understanding the relationship between algebraic Bergman kernels and the finite type of boundaries in complex domains.
method Analyzing algebraic Bergman kernels and their implications on the finite type of boundaries in smoothly bounded pseudoconvex domains in C2. result The boundary of a smoothly bounded pseudoconvex domain with an algebraic Bergman kernel of degree d is of finite type with type r≤2d. Enhances KLR for indefinite kernels with L1-norm regularization.
problem Classifying with indefinite kernels captures more domain-specific information.
method Introduces L1-norm regularization to induce sparsity and a proximal linearized algorithm. result Superior performance in accuracy and sparsity on multiple datasets.
The Bergman kernel's minimal point determines domain properties.
problem Characterizing domains based on the Bergman kernel's minimal point.
method Using Suita conjecture and properties of Bergman and Szegö kernels.
result The Bergman kernel's minimal point identifies specific domain types.
Study improves multi-class domain generalization with a new error bound.
problem Improving multi-class classification performance across multiple domains.
method Kernel-based learning algorithm with a logarithmic generalization error bound.
result Achieved significant performance gains over a pooling strategy empirically.
Paper introduces a new Poisson kernel for strongly pseudoconvex domains.
problem Developing a new mathematical tool for strongly pseudoconvex domains.
method Introducing a maximal plurisubharmonic function called the pluricomplex Poisson kernel.
result The pluricomplex Poisson kernel shares properties with the classical Poisson kernel and reproduces pluriharmonic functions.
Inspired by the work of Z. Lu and G. Tian [21] in the compact setting, in this paper we address the problem of studying the Szegö kernel of the disk bundle over a noncompact Kähler manifold. In particular we compute the Szegö kernel of the disk bundle over a Cartan-Hartogs domain based on a bounded symmetric domain. Th…
Advanced kernels improve Gaussian process accuracy by incorporating domain knowledge.
problem Improving function approximation accuracy in Gaussian processes.
method Advanced kernel designs that enforce specific function properties (symmetry, periodicity) and non-stationarity.
result Advanced kernels significantly enhance function approximation accuracy and relevance.
G-FuNK learns solutions for nonlinear PDEs on multiple domains and parameters.
problem Predicting time-dependent dynamics of complex systems governed by nonlinear PDEs with varying parameters and domains.
method Graph Fourier Neural Kernels combining domain-adapted and transferable components for non-diffusive and diffusive terms.
result G-FuNK achieves low relative errors on unseen domains and fiber fields, significantly accelerating predictions.
A statistical test of independence may be constructed using the Hilbert-Schmidt Independence Criterion (HSIC) as a test statistic. The HSIC is defined as the distance between the embedding of the joint distribution, and the embedding of the product of the marginals, in a Reproducing Kernel Hilbert Space (RKHS). It has …
Study on expressive power of Euclidean kernels and efficient kernel learning.
problem Limiting the expressive power of kernel methods and improving kernel learning efficiency.
method Define Euclidean kernels, analyze their geometric and spectral properties, and develop efficient algorithms for kernel learning.
result Prove limitations on the expressive power of Euclidean kernels and derive efficient algorithms for kernel learning.
Graph kernels assess graph similarity for various applications.
problem Assessing similarity between graphs for predictions.
method Review and comparison of existing graph kernels.
result State-of-the-art graph kernels reviewed and compared.
New algorithm clusters data and learns kernels without relaxing constraints.
problem Learning kernels or distance metrics from pairwise constraints without losing generalization.
method Joint clustering and kernel learning without relaxing constraints.
result Outperforms existing approaches on diverse datasets.
Improved guarantees for misspecified kernelized bandit optimization.
problem Misspecification in kernelized bandit optimization.
method Localization and domain splitting techniques.
result Logarithmic or polylogarithmic growth of misspecification amplification.
TGG improves zero-shot and few-shot learning by explicitly modeling and utilizing seen-unseen domain relations.
problem Lack of data in unseen domains hinders generalization in zero-shot and few-shot learning.
method TGG generates explicit instance-level graphs to model and utilize seen-unseen domain relations, addressing domain shift.
result TGG outperforms existing methods in zero-shot, generalized zero-shot, and few-shot learning.
New GL-GP models learn covariance respecting domain geometry.
problem Suboptimal results from nonparametric regression on restricted domains.
method Graph Laplacian based Gaussian Processes (GL-GPs) with Nyström extension.
result Performance gains in various applications.
A new kernel for multi-output Gaussian processes reduces undesirable scale effects.
problem Predicting multiple output variables simultaneously with Gaussian processes.
method Design a new kernel (MOCSM) using convolution in the spectral domain to model cross channel dependencies.
result MOCSM kernel reduces undesirable scale effects compared to the Multi-Output Spectral Mixture kernel.
Paper tackles domain generalization by minimizing domain-based covariance.
problem Training data and test data have different distributions, leading to poor generalization.
method Find a central subspace minimizing domain-based covariance while preserving functional relationships.
result The proposed method achieves better generalization performance on unseen test datasets.
This work studies nonnegativity-preserving kernels for stochastic equations and their applications.
problem Nonnegativity preservation in stochastic Volterra equations and related processes.
method Characterization and application of completely monotone kernels; approximation schemes for weak error.
result Positive linear combinations of decaying exponentials can be used for second-order approximation schemes.
We consider the Hypothesis Transfer Learning (HTL) problem where one incorporates a hypothesis trained on the source domain into the learning procedure of the target domain. Existing theoretical analysis either only studies specific algorithms or only presents upper bounds on the generalization error but not on the exc…
This paper investigates domain generalization: How to take knowledge acquired from an arbitrary number of related domains and apply it to previously unseen domains? We propose Domain-Invariant Component Analysis (DICA), a kernel-based optimization algorithm that learns an invariant transformation by minimizing the diss…
New kernels capture both local and non-local interactions efficiently.
problem Designing kernels that capture both local and non-local interactions while remaining computationally tractable.
method Spectral truncation kernels based on C∗-algebra. result Spectral truncation kernels induce interactions across the data function domain and reduce computational cost.
We introduce a kernel method for manifold alignment (KEMA) and domain adaptation that can match an arbitrary number of data sources without needing corresponding pairs, just few labeled examples in all domains. KEMA has interesting properties: 1) it generalizes other manifold alignment methods, 2) it can align manifold…
Kernelmethods library simplifies kernel-based ML in Python.
problem Lack of a well-defined Python library for kernel-based ML.
method Develops a Python library that supports various kernel types and customizations.
result Enables efficient, modular, and interoperable kernel-based operations.
Two-dimensional domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.
problem Characterizing domains with Kähler-Einstein Bergman metrics.
method Asymptotics of derivatives of the Bergman kernel along critically tangent paths.
result Two-dimensional pseudoconvex domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.
Inspired by Berndtsson's work on the subharmonicity property of the Bergman kernel, we give a local variation formula of the full Bergman kernels associated to deformations of complex manifolds. In compact case, it follows from the reproducing property of the Bergman kernel and the curvature formula of the 0-th direct …
New wavelet frames constructed from reproducing kernels for continuous and discrete domains.
problem Generating wavelet frames on non-Euclidean structures.
method Spectral filtering of integral operators associated with reproducing kernels.
result Discrete frames as Monte Carlo estimates of continuous frames, with finite-sample rates derived.
Characterizes kernel of linearization for minimal surfaces problem
problem Characterizing kernel of linearization for minimal surfaces problem
method Show kernel consists of potential fields and TT fields
result In whole-space Euclidean decomposition, kernel consists of potential fields and TT fields
New kernel HMK improves Gaussian process expressiveness and supports harmonizable covariances.
problem Improving the expressiveness of Gaussian processes with non-stationary kernels.
method Proposed harmonizable mixture kernel (HMK) and variational Fourier features.
result HMK interpolates between local patterns and offers robust kernel learning.
The paper develops a multi-kernel method with sparsity constraint for regression.
problem Developing a robust regression method with sparsity constraints.
method Banach-space formulation, generalized total-variation regularization, multi-kernel expansion, adaptive kernel positions, ℓ1 penalty on coefficients. result The method achieves sparsity in the kernel coefficients, reducing the number of active kernels to the number of data points.
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.
We demonstrate that distributed block coordinate descent can quickly solve kernel regression and classification problems with millions of data points. Armed with this capability, we conduct a thorough comparison between the full kernel, the Nyström method, and random features on three large classification tasks from va…
We propose a class of intrinsic Gaussian processes (in-GPs) for interpolation, regression and classification on manifolds with a primary focus on complex constrained domains or irregular shaped spaces arising as subsets or submanifolds of R, R2, R3 and beyond. For example, in-GPs can accommodate spatial domains arising…
Quantitative Sobolev extensions lead to Neumann heat kernel bounds.
problem Bounding Neumann heat kernels for domains with integral Ricci curvature.
method Quantitative Sobolev extension operators and Neumann heat kernel estimates.
result Uniform bounds on Neumann heat kernels and eigenvalues.
Boundary effects inflate variance in Gaussian processes, leading to acquisition bias.
problem Boundary-induced acquisition bias in Gaussian processes.
method Traced root cause to geometric mechanism of kernel truncation at domain boundaries.
result Boundary effects create distortion that worsens with dimensionality, affecting acquisition behavior.
Novel boundary integral equations for Dirac operators in 3D Lipschitz domains.
problem Developing equations for Dirac operators in complex 3D domains.
method First-kind boundary integral equations, generalized Garding inequalities, Fredholm operators, finite dimensional kernels, Betti numbers.
result Finite dimensional kernels equal to the sum of Betti numbers, explaining the bilinear forms.
The Bergman-Szegő kernel is analyzed for weakly pseudoconvex CR manifolds of finite type.
problem Analyzing the Bergman-Szegő kernel for specific CR manifolds.
method Constructing a parametrix for the Szegő kernel, extending earlier results.
result Extending Fefferman's boundary asymptotics to weakly pseudoconvex domains in \(\mathbb{C}^{2}\).
New BdryMatérn GP model for reliable boundary integration on irregular domains.
problem Incorporating boundary information in Gaussian process models for complex phenomena.
method Proposes a novel BdryMatérn GP framework with a new covariance kernel derived via path integral and stochastic PDE.
result Sample paths from the BdryMatérn GP satisfy desired boundaries with smoothness control on derivatives.
SOBER framework optimizes Bayesian optimization tasks efficiently.
problem Challenges in parallel Bayesian optimization.
method Probabilistic Lifting with Kernel Quadrature.
result Versatile and flexible batch Bayesian optimization.
GPs with neural network dual kernels improve reinforcement learning performance.
problem Combining the strengths of DNNs and GPs for reinforcement learning.
method Apply GPs with neural network dual kernels to solve reinforcement learning tasks.
result GPs with neural network dual kernels perform at least as well as conventional methods on the mountain-car problem.
3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.
problem Understanding which 3-manifold groups can have convex co-compact representations.
method Analyzing representations of 3-manifold groups into projective general linear group, focusing on convex co-compactness.
result Fundamental groups of closed irreducible orientable 3-manifolds can only admit convex co-compact representations if they are geometric or hyperbolic.