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About 2,645 papers for “Reproducing kernel Hilbert space”

Reproducing Kernel Hilbert Space vs. Frame Estimates

Palle E. T. Jørgensen, Myung-Sin Song · Mathematics · 2015

We consider conditions on a given system F of vectors in Hilbert space H, forming a frame, which turn H into a reproducing kernel Hilbert space. It is assumed that the vectors in F are functions on some set Ω . We then identify conditions on these functions which automatically give H the structure of a reproducing kernel Hilbert space of functions on Ω. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes.

Cited by 2Open access

Policy Search in Reproducing Kernel Hilbert Space

Ngo, Vien Anh, Englert, Peter, Toussaint, Marc · Research Portal (Queen's University Belfast) · 2016

Modeling policies in reproducing kernel Hilbert space (RKHS) renders policy gradient reinforcement learning algorithms non-parametric. As a result, the policies become very flexible and have a rich representational potential without a pre-defined set of features. However, their performances might be either non-covariant under re-parameterization of the chosen kernel, or very sensitive to step-size selection. In this paper, we propose to use a general framework to derive a new RKHS policy search technique. The new derivation leads to both a natural RKHS actor-critic algorithm and a RKHS expectation maximization (EM) policy search algorithm. Further,we show that kernelization enables us to learn in partially observable (POMDP) tasks which is considered daunting for parametric approaches. Via sparsification, a small set of "support vectors" representing the history is shown to be effectively discovered.For evaluations, we use three simulated (PO)MDP reinforcement learning tasks, and a simulated PR2's robotic manipulation task. The results demonstrate the effectiveness of the new RKHS policy search framework in comparison to plain RKHS actor-critic, episodic natural actor-critic, plain actor-critic, and PoWER approaches.

Cited by 0

Robust Estimation in Reproducing Kernel Hilbert Space

Joseph A. Gallego-Mejia, Fabio A. González · 2019

Our work shows that estimating the mean in a feature space induced by certain kinds of kernels is the same as doing a robust mean estimation using an M-estimator in the original problem space. In particular, we show that calculating the average on a feature space induced by a Gaussian kernel is equivalent to perform robust mean estimation with the Welsch M-estimator. Besides, a new framework is proposed that was used to build four new robust kernels: Tukey’s, Andrews’, Huber’s and Cauchy’s robust kernels. The new robust kernels, combined with kernel matrix factorization clustering algorithm, were compared to state-of-the-art algorithms in clustering tasks. The result shows that some of the new robust kernels perform in a par with state-of-the-art algorithms.

Cited by 0Open access

Subspace Classifier in Reproducing Kernel Hilbert Space

Koji Tsuda Electrotechnical, Koji Tsuda · 1999

To improve the performance of subspace classifier, it is effective to reduce the dimensionality of the intersections between subspaces. For this purpose, the feature space is mapped implicitly to a high dimensional reproducing kernel Hilbert space and the subspace classifier is applied in this space. As a result of Hiragana recognition experiment, our classifier outperformed the conventional subspace classifier. 1 Introduction Subspace classifier is the pattern recognition method which uses a linear subspace for describing a class[1]. In training, a subspace is fit to training samples so that the sum of the squared distances between the samples and the subspace is minimized. In classifying an unlabeled sample, the sample is classified to the class whose subspace is the nearest. It is known that the performance of subspace classifier is affected by the dimensionality of the intersections of subspaces[1]. When the dimensionality of the feature space is larger than the number of traini...

Cited by 0

Kernel Principal Component Regression in Reproducing Kernel Hilbert Space

Chooleewan Dachapak, Shunshoku Kanae, Zi-Jiang Yang et al. · Proceedings of the ISCIE International Symposium on Stochastic Systems Theory and its Applications · 2003

In this study, we proposed Kernel Principal Component Analysis (KPCA) which is applied for feature selection in a high-dimensional feature space which is nonlinearly mapped from an input space by a Gaussian kernel function. By using Mercer Kernels, we can compute principal components in a high dimensional feature space. Then, the extracted features are employed as preprocessing step for an ordinary least squares regression in the feature space which is Reproducing Kernel Hilbert Space (RKHS).

Cited by 2Open access

Quantum Policy Gradient in Reproducing Kernel Hilbert Space

David M. Bossens, Kishor Bharti, Jayne Thompson · arXiv (Cornell University) · 2024

Parametrised quantum circuits offer expressive and data-efficient representations for machine learning. Due to quantum states residing in a high-dimensional Hilbert space, parametrised quantum circuits have a natural interpretation in terms of kernel methods. The representation of quantum circuits in terms of quantum kernels has been studied widely in quantum supervised learning, but has been overlooked in the context of quantum RL. This paper proposes the use of kernel policies and quantum policy gradient algorithms for quantum-accessible environments. After discussing the properties of such policies and a demonstration of classical policy gradient on a coherent policy in a quantum environment, we propose parametric and non-parametric policy gradient and actor-critic algorithms with quantum kernel policies in quantum environments. This approach, implemented with both numerical and analytical quantum policy gradient techniques, allows exploiting the many advantages of kernel methods, including data-driven forms for functions (and their gradients) as well as tunable expressiveness. The proposed approach is suitable for vector-valued action spaces and each of the formulations demonstrates a quadratic reduction in query complexity compared to their classical counterparts. We propose actor-critic algorithms based on stochastic policy gradient, deterministic policy gradient, and natural policy gradient, and demonstrate additional query complexity reductions compared to quantum policy gradient algorithms under favourable conditions.

Cited by 0Open access

Step size adaptation in reproducing kernel Hilbert space

Nicol N. Schraudolph, Alex J. Smola, Thorsten Joachims · 2006

This paper presents an online support vector machine (SVM) that uses the stochastic meta-descent (SMD) algorithm to adapt its step size automatically. We formulate the online learning problem as a stochastic gradient descent in reproducing kernel Hilbert space (RKHS) and translate SMD to the nonparametric setting, where its gradient trace parameter is no longer a coefficient vector but an element of the RKHS. We derive efficient updates that allow us to perform the step size adaptation in linear time. We apply the online SVM framework to a variety of loss functions, and in particular show how to handle structured output spaces and achieve efficient online multiclass classification. Experiments show that our algorithm outperforms more primitive methods for setting the gradient step size.

Cited by 0

KERE: Expectile Regression in Reproducing Kernel Hilbert Space

Yi Yang, Teng Zhang, Hui Zou · 2015

An efficient algorithm inspired by majorization-minimization principle for solving the entire solution path of a flexible nonparametric expectile regression estimator constructed in a reproducing kernel Hilbert space.

Cited by 0Open access

A Kernel Affine Projection-Like Algorithm in Reproducing Kernel Hilbert Space

Qishuai Wu, Yingsong Li, Yuriy Zakharov et al. · IEEE Transactions on Circuits & Systems II Express Briefs · 2019

A kernel affine projection-like algorithm (KAPLA) is proposed in reproducing kernel Hilbert space in non-Gaussian environments. The cost function for the developed algorithm is constructed by using the correntropy approach and Gaussian kernel to deal with nonlinear channel estimation. The devised algorithm can efficiently operate in the impulse noise. As a consequence, the proposed KAPLA algorithm provides good performance for nonlinear channel equalization in impluse-noise environments. Simulations results in different mixed noise environments verify the superior behavior of KAPLA compared to known algorithms.

Cited by 40

Kernel Center Adaptation in the Reproducing Kernel Hilbert Space Embedding Method

Sai Tej Paruchuri, Jia Guo, Andrew J. Kurdila · arXiv (Cornell University) · 2020

The performance of adaptive estimators that employ embedding in reproducing kernel Hilbert spaces (RKHS) depends on the choice of the location of basis kernel centers. Parameter convergence and error approximation rates depend on where and how the kernel centers are distributed in the state-space. In this paper, we develop the theory that relates parameter convergence and approximation rates to the position of kernel centers. We develop criteria for choosing kernel centers in a specific class of systems - ones in which the state trajectory regularly visits the neighborhood of the positive limit set. Two algorithms, based on centroidal Voronoi tessellations and Kohonen self-organizing maps, are derived to choose kernel centers in the RKHS embedding method. Finally, we implement these methods on two practical examples and test their effectiveness.

Cited by 0Open access

Kernel VICReg for Self-Supervised Learning in Reproducing Kernel Hilbert Space

M. Hadi Sepanj, Benyamin Ghojogh, Saed Moradi et al. · Big Data and Cognitive Computing · 2026

Self-supervised learning (SSL) has emerged as a powerful paradigm for representation learning by optimizing geometric objectives, such as invariance to augmentations, variance preservation, and feature decorrelation, without requiring labels. However, most existing methods operate in Euclidean space, limiting their ability to capture nonlinear dependencies and geometric structures. In this work, we propose Kernel VICReg, a novel self-supervised learning framework that pulls the VICReg objective into a Reproducing Kernel Hilbert Space (RKHS). By kernelizing each term of the loss, variance, invariance, and covariance, we obtain a general formulation that operates on double-centered kernel matrices and Hilbert–Schmidt norms, enabling nonlinear feature learning without explicit mappings. We demonstrate that Kernel VICReg mitigates the risk of representational collapse under challenging conditions and improves performance on datasets exhibiting nonlinear structure or limited sample regimes. Empirical evaluations across MNIST, CIFAR-10, STL-10, TinyImageNet, and ImageNet100 show consistent gains over Euclidean VICReg, with particularly strong improvements on datasets where nonlinear structures are prominent. UMAP visualizations are provided only as a qualitative illustration of embedding geometry and are not used as a calibration or statistical validation. Our results suggest that kernelizing SSL objectives is a promising direction for bridging classical kernel methods with modern representation learning.

Cited by 0Open access