PGF kernels analyze spherical data using generalized RBF kernels.
problem Analysis of spherical data.
method Introduced PGF kernels and a semi-parametric learning algorithm.
result PGF kernels generalize RBF kernels for spherical data.
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48 results for “Kernel Function”
Extends Tanimoto kernel to real-valued functions.
problem Measuring similarity between real-valued functions.
method Unified representation of real-valued functions via sets, derived general form of the kernel, explicit feature representation, and smooth approximation.
result General Tanimoto kernel for real-valued functions.
We develop Fourier methods to expand translation-invariant kernels.
problem Constructing orthonormal expansions for translation-invariant kernels.
method Fourier analytic technique to derive explicit expansions.
result Explicit expansions for various kernels (Matérn, Cauchy, Gaussian).
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 -algebra.
result Spectral truncation kernels induce interactions across the data function domain and reduce computational cost.
MKLpy simplifies Multiple Kernel Learning in Python.
problem Learning optimal kernel functions from data.
method Python-based framework for Multiple Kernel Learning algorithms.
result Maximizes usability and simplifies development of novel solutions.
AEN-RBF kernel improves robustness in Bayesian optimization for complex systems.
problem Bayesian optimization struggles with outliers in RBF kernel, leading to poor performance.
method Proposes AEN-RBF kernel function, demonstrating improved robustness and convergence.
result The AEN-RBF kernel function reduces mean squared prediction error and improves convergence.
Gaussian processes are flexible function approximators, with inductive biases controlled by a covariance kernel. Learning the kernel is the key to representation learning and strong predictive performance. In this paper, we develop functional kernel learning (FKL) to directly infer functional posteriors over kernels. I…
RKUM is an R package for robust kernel-based unsupervised methods.
problem Robust analysis under contaminated or noisy data conditions.
method Robust kernel covariance and cross-covariance operators using generalized loss functions.
result RKUM reduces sensitivity to contamination and effectively identifies outliers.
Neural network learns kernel functions for survival analysis and prediction intervals.
problem Predicting survival times for individuals based on similar training subjects.
method Develops a neural network framework to learn kernel functions for kernel survival analysis and uses these to construct valid prediction intervals.
result Neural network survival estimators are competitive with existing methods and provide valid prediction intervals.
To the best of our knowledge, there are no general well-founded robust methods for statistical unsupervised learning. Most of the unsupervised methods explicitly or implicitly depend on the kernel covariance operator (kernel CO) or kernel cross-covariance operator (kernel CCO). They are sensitive to contaminated data, …
Many unsupervised kernel methods rely on the estimation of the kernel covariance operator (kernel CO) or kernel cross-covariance operator (kernel CCO). Both kernel CO and kernel CCO are sensitive to contaminated data, even when bounded positive definite kernels are used. To the best of our knowledge, there are few well…
The study explains how neural networks align their kernels to target functions during training.
problem Understanding how neural networks align their kernels to target functions during training.
method Theoretical analysis of kernel evolution in toy models and deep networks.
result Kernel alignment naturally emerges during training to accelerate convergence and improve generalization.
Kernel-based function approximation improves reinforcement learning performance.
problem Average reward reinforcement learning in infinite horizon settings.
method Optimistic algorithm based on kernel ridge regression.
result No-regret performance guarantees and confidence intervals for kernel-based predictions.
Adaptive RBF-KAN: A Comparative Evaluation of Dynamic Shape Parameters in Kolmogorov-Arnold Networksstat.ML
Adaptive RBF-KAN improves KANs by dynamically adjusting kernel parameters.
problem Efficiently approximating multivariate functions using learnable univariate edge functions.
method Integrates LOOCV-based kernel scale estimation with adaptive kernel learning.
result Adaptive RBF-KAN outperforms fixed kernel KANs on various benchmark functions.
We develop a multi-kernel based regression method for graph signal processing where the target signal is assumed to be smooth over a graph. In multi-kernel regression, an effective kernel function is expressed as a linear combination of many basis kernel functions. We estimate the linear weights to learn the effective …
Generalized Spectral Kernelsstat.ML
In this paper we propose a family of tractable kernels that is dense in the family of bounded positive semi-definite functions (i.e. can approximate any bounded kernel with arbitrary precision). We start by discussing the case of stationary kernels, and propose a family of spectral kernels that extends existing approac…
Kernel methods' derivatives make complex models more interpretable.
problem Interpreting complex kernel models.
method Deriving kernel functions' derivatives and applying them to various kernel methods.
result Derivatives of kernel functions can be computed and applied to improve model interpretation.
Unified proofs of weak holomorphic Morse inequalities using Bergman kernel functions.
problem Weak holomorphic Morse inequalities on various types of manifolds.
method Asymptotic Bergman kernel functions and Bochner-Kodaira-Nakano formulas.
result Unified proofs of weak holomorphic Morse inequalities.
This paper explains why ResNets generalize better than FFNets using neural tangent kernels.
problem Understanding why deep ResNets generalize better than deep FFNets.
method Using neural tangent kernels to compare the learnability of functions induced by the kernels of ResNets and FFNets.
result The kernel of ResNets does not exhibit degeneracy as depth increases, unlike FFNets.
Given only information in the form of similarity triplets "Object A is more similar to object B than to object C" about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a low-dimensional Euclidean embedding of the data set that reflects the given similar…
Paper characterizes embeddability of function spaces into -type RKBS via metric entropy.
problem Characterizing embeddability of function spaces into -type RKBS.
method Establishes a connection between metric entropy growth and embeddability.
result A bound on metric entropy growth allows embedding into -type RKBS.
Kernel methods are among the most popular techniques in machine learning. From a frequentist/discriminative perspective they play a central role in regularization theory as they provide a natural choice for the hypotheses space and the regularization functional through the notion of reproducing kernel Hilbert spaces. F…
Develops interpretable low-dimensional kernels with conic discriminant functions.
problem Improving interpretability in kernel-based classification models.
method Gradually constructs simple feature maps leading to interpretable low-dimensional kernels.
result Obtains high accuracy results without extensive hyperparameter tuning.
Optimal kernel in KR can be data-dependent, improving model performance.
problem Fixed kernel in KR limits model performance.
method Considered data-dependent kernels for KR, using posterior covariance.
result Data-dependent kernel choice leads to optimal performance.
Learning "best" kernels from data in Gaussian process regression. With application to aerodynamicsstat.ML
Paper learns optimal kernels for Gaussian process regression in aerodynamics.
problem Approximating complex functions from limited data in aerodynamics.
method Two algorithms: Kernel Flow and Spectral Kernel Ridge Regression.
result Explicit construction of optimal kernels based on target function features.
We analyze how kernel machines and neural networks learn different frequency modes of the target function as data size increases.
problem Understanding how kernel machines and neural networks learn different frequency modes of the target function as data size increases.
method Theoretical methods from Gaussian processes and statistical physics, combined with simulations on synthetic data and MNIST dataset.
result Kernel machines and neural networks fit successively higher spectral modes of the target function as the size of the training set grows.
As a novel similarity measure that is defined as the expectation of a kernel function between two random variables, correntropy has been successfully applied in robust machine learning and signal processing to combat large outliers. The kernel function in correntropy is usually a zero-mean Gaussian kernel. In a recent …
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.
Standard kernels such as Matérn or RBF kernels only encode simple monotonic dependencies within the input space. Spectral mixture kernels have been proposed as general-purpose, flexible kernels for learning and discovering more complicated patterns in the data. Spectral mixture kernels have recently been generalized in…
Bayesian kernel regression improves functional output prediction.
problem Functional output regression in supervised learning.
method Kernel methods, leveraging covariance structure within function values.
result Enhanced prediction accuracy and handling of high-dimensional nonlinearity.
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.
The paper introduces new estimators for multivariate functions using Fourier methods.
problem Estimating multivariate functions like densities and regression functions.
method Monte Carlo estimators based on the Fourier integral theorem.
result Established rates of convergence for new estimators, often superior to existing methods.
New kernels from ELU and GELU networks reveal non-trivial fixed points.
problem Understanding fixed-point dynamics in deep neural networks with ELU and GELU activations.
method Deriving covariance functions and analyzing fixed-point dynamics of ELU and GELU networks.
result ELU and GELU networks exhibit non-trivial fixed-point dynamics, explaining implicit regularization in overparameterized models.
Prominently used in support vector machines and logistic regressions, kernel functions (kernels) can implicitly map data points into high dimensional spaces and make it easier to learn complex decision boundaries. In this work, by replacing the inner product function in the softmax layer, we explore the use of kernels …
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 propose a representation of graph as a functional object derived from the power iteration of the underlying adjacency matrix. The proposed functional representation is a graph invariant, i.e., the functional remains unchanged under any reordering of the vertices. This property eliminates the difficulty of handling e…
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.
Kernel for STL formulae enables machine learning in temporal logic.
problem Lack of a kernel for STL formulae.
method Define a kernel for STL formulae and embed them into a Hilbert space.
result Kernel-based machine learning algorithms can now be applied to STL formulae.
Kernel methods are an extremely popular set of techniques used for many important machine learning and data analysis applications. In addition to having good practical performances, these methods are supported by a well-developed theory. Kernel methods use an implicit mapping of the input data into a high dimensional f…
Positive-definite kernel functions are fundamental elements of kernel methods and Gaussian processes. A well-known construction of such functions comes from Bochner's characterization, which connects a positive-definite function with a probability distribution. Another construction, which appears to have attracted less…
Positive definite operator-valued kernels generalize the well-known notion of reproducing kernels, and are naturally adapted to multi-output learning situations. This paper addresses the problem of learning a finite linear combination of infinite-dimensional operator-valued kernels which are suitable for extending func…
Paper introduces a new kernel model for PSD-valued functions with theoretical guarantees and applications.
problem Enforcing positive semi-definiteness (PSD) in function models with good performance and theoretical guarantees.
method Kernel sum-of-squares model for PSD-valued functions, extending previous models for non-negative scalar functions.
result The model constitutes a universal approximator of PSD functions and can represent any smooth and strongly convex function.
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.
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.
This paper introduces the concept of kernels on fuzzy sets as a similarity measure for -valued functions, a.k.a. \emph{membership functions of fuzzy sets}. We defined the following classes of kernels: the cross product, the intersection, the non-singleton and the distance-based kernels on fuzzy sets. Applicabili…
We show that under very general assumptions the partial Bergman kernel function of sections vanishing along an analytic hypersurface has exponential decay in a neighborhood of the vanishing locus. Considering an ample line bundle, we obtain a uniform estimate of the Bergman kernel function associated to a singular metr…
Sparse Gaussian processes with compact kernels for faster inference.
problem Efficient Gaussian process inference with high computational complexity.
method Parametric families of compactly-supported kernels for sparse matrix representations.
result Sub-quadratic inference complexity and improved performance on real-world tasks.
Explicit formula for Bergman kernel of abelian varieties proved.
problem Explicit formula for Bergman kernel of polarized abelian varieties.
method Explicit formula for Bergman kernel of polarized abelian varieties.
result Explicit formula for Bergman kernel of polarized abelian varieties.