Given a reproducing kernel Hilbert space H of real-valued functions and a suitable measure mu over the source space D (subset of R), we decompose H as the sum of a subspace of centered functions for mu and its orthogonal in H. This decomposition leads to a special case of ANOVA kernels, for which the functional ANOVA representation of the best predictor can be elegantly derived, either in an interpolation or regularization framework. The proposed kernels appear to be particularly convenient for analyzing the e ffect of each (group of) variable(s) and computing sensitivity indices without recursivity.
Regularized approaches have been successfully applied to linear system identification in recent years. Many of them model unknown impulse responses exploiting the so called Reproducing Kernel Hilbert spaces (RKHSs) that enjoy the notable property of being in one-to-one correspondence with the class of positive semidefinite kernels. The necessary and sufficient condition for a RKHS to be stable, i.e. to contain only BIBO stable linear dynamic systems, has been known in the literature at least since 2006. However, an open question still persists and concerns the equivalence of such condition wit…
We consider the problem of optimising functions in the reproducing kernel Hilbert space (RKHS) of a Matérn kernel with smoothness parameter $ν$ over the domain $[0,1]^d$ under noisy bandit feedback. Our contribution, the $π$-GP-UCB algorithm, is the first practical approach with guaranteed sublinear regret for all $ν>1$ and $d \geq 1$. Empirical validation suggests better performance and drastically improved computational scalablity compared with its predecessor, Improved GP-UCB.
KU Leuven（鲁汶大学）；Fudan University（复旦大学）；Shanghai Jiao Tong University（上海交通大学）
This paper generalizes regularized regression problems in a hyper-reproducing kernel Hilbert space (hyper-RKHS), illustrates its utility for kernel learning and out-of-sample extensions, and proves asymptotic convergence results for the introduced regression models in an approximation theory view. Algorithmically, we consider two regularized regression models with bivariate forms in this space, including kernel ridge regression (KRR) and support vector regression (SVR) endowed with hyper-RKHS, and further combine divide-and-conquer with Nyström approximation for scalability in large sample cas…
Simons Institute for the Theory of Computing, University of California, Berkeley（加州大学伯克利分校西蒙斯理论计算研究所）；Department of Statistics and Data Science, Yale University（耶鲁大学统计与数据科学系）
We prove that the reproducing kernel Hilbert spaces (RKHS) of a deep neural tangent kernel and the Laplace kernel include the same set of functions, when both kernels are restricted to the sphere $\mathbb{S}^{d-1}$. Additionally, we prove that the exponential power kernel with a smaller power (making the kernel less smooth) leads to a larger RKHS, when it is restricted to the sphere $\mathbb{S}^{d-1}$ and when it is defined on the entire $\mathbb{R}^d$.
Reproducing kernel Hilbert spaces (RKHSs) are key spaces for machine learning that are becoming popular also for linear system identification. In particular, the so-called stable RKHSs can be used to model absolutely summable impulse responses. In combination e.g. with regularized least squares they can then be used to reconstruct dynamic systems from input-output data. In this paper we provide new structural properties of stable RKHSs. The relation between stable kernels and other fundamental classes, like those containing absolutely summable or finite-trace kernels, is elucidated. These insi…
We present DARTR: a Data Adaptive RKHS Tikhonov Regularization method for the linear inverse problem of nonparametric learning of function parameters in operators. A key ingredient is a system intrinsic data-adaptive (SIDA) RKHS, whose norm restricts the learning to take place in the function space of identifiability. DARTR utilizes this norm and selects the regularization parameter by the L-curve method. We illustrate its performance in examples including integral operators, nonlinear operators and nonlocal operators with discrete synthetic data. Numerical results show that DARTR leads to an …
Kernel methods are powerful tools to capture nonlinear patterns behind data. They implicitly learn high (even infinite) dimensional nonlinear features in the Reproducing Kernel Hilbert Space (RKHS) while making the computation tractable by leveraging the kernel trick. Classic kernel methods learn a single layer of nonlinear features, whose representational power may be limited. Motivated by recent success of deep neural networks (DNNs) that learn multi-layer hierarchical representations, we propose a Stacked Kernel Network (SKN) that learns a hierarchy of RKHS-based nonlinear features. SKN int…
University of Oxford（牛津大学）；Microsoft Research Cambridge（微软剑桥研究院）
Feature attribution for kernel methods is often heuristic and not individualised for each prediction. To address this, we turn to the concept of Shapley values~(SV), a coalition game theoretical framework that has previously been applied to different machine learning model interpretation tasks, such as linear models, tree ensembles and deep networks. By analysing SVs from a functional perspective, we propose \textsc{RKHS-SHAP}, an attribution method for kernel machines that can efficiently compute both \emph{Interventional} and \emph{Observational Shapley values} using kernel mean embeddings o…
We study quadrature rules for functions from an RKHS, using nodes sampled from a determinantal point process (DPP). DPPs are parametrized by a kernel, and we use a truncated and saturated version of the RKHS kernel. This link between the two kernels, along with DPP machinery, leads to relatively tight bounds on the quadrature error, that depends on the spectrum of the RKHS kernel. Finally, we experimentally compare DPPs to existing kernel-based quadratures such as herding, Bayesian quadrature, or leverage score sampling. Numerical results confirm the interest of DPPs, and even suggest faster r…
Universita’ degli Studi di Genova（热那亚大学）；University of Stuttgart（斯图加特大学）；Massachusetts Institute of Technology（麻省理工学院）；Istituto Italiano di Tecnologia（意大利理工学院）
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 a number of detailed examples.
McGill University, Department of Mathematics and Statistics（麦吉尔大学数学与统计学系）；Université Marie et Louis Pasteur, CNRS, LmB (UMR 6623)（玛丽·路易斯·帕斯特大学、法国国家科学研究中心、LmB（UMR 6623））
Kernel methods have been among the most popular techniques in machine learning, where learning tasks are solved using the property of reproducing kernel Hilbert space (RKHS). In this paper, we propose a novel data analysis framework with reproducing kernel Hilbert $C^*$-module (RKHM), which is another generalization of RKHS than vector-valued RKHS (vv-RKHS). Analysis with RKHMs enables us to deal with structures among variables more explicitly than vv-RKHS. We show the theoretical validity for the construction of orthonormal systems in Hilbert $C^*$-modules, and derive concrete procedures for …
NTT Network Service Systems Laboratories, NTT Corporation（NTT网络服务系统实验室，NTT公司）；Graduate School of Science and Technology, Keio University（庆应义塾大学科学与技术研究生院）；Center for Data Science, Ehime University（爱媛大学数据科学中心）；Center for Advanced Intelligence Project, RIKEN（理化学研究所高级智能项目中心）；Faculty of Science and Technology, Keio University（庆应义塾大学科学与技术学部）
Kernel methods have been among the most popular techniques in machine learning, where learning tasks are solved using the property of reproducing kernel Hilbert space (RKHS). In this paper, we propose a novel data analysis framework with reproducing kernel Hilbert $C^*$-module (RKHM) and kernel mean embedding (KME) in RKHM. Since RKHM contains richer information than RKHS or vector-valued RKHS (vvRKHS), analysis with RKHM enables us to capture and extract structural properties in such as functional data. We show a branch of theories for RKHM to apply to data analysis, including the representer…
We propose a generic spatiotemporal event forecasting method, which we developed for the National Institute of Justice's (NIJ) Real-Time Crime Forecasting Challenge. Our method is a spatiotemporal forecasting model combining scalable randomized Reproducing Kernel Hilbert Space (RKHS) methods for approximating Gaussian processes with autoregressive smoothing kernels in a regularized supervised learning framework. While the smoothing kernels capture the two main approaches in current use in the field of crime forecasting, kernel density estimation (KDE) and self-exciting point process (SEPP) mod…
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
This work studies the convergence and finite sample approximations of entropic regularized Wasserstein distances in the Hilbert space setting. Our first main result is that for Gaussian measures on an infinite-dimensional Hilbert space, convergence in the 2-Sinkhorn divergence is {\it strictly weaker} than convergence in the exact 2-Wasserstein distance. Specifically, a sequence of centered Gaussian measures converges in the 2-Sinkhorn divergence if the corresponding covariance operators converge in the Hilbert-Schmidt norm. This is in contrast to the previous known result that a sequence of c…
Global sensitivity analysis is the main quantitative technique for identifying the most influential input variables in a numerical simulation model. In particular when the inputs are independent, Sobol' sensitivity indices attribute a portion of the output of interest variance to each input and all possible interactions in the model, thanks to a functional ANOVA decomposition. On the other hand, moment-independent sensitivity indices focus on the impact of input variables on the whole output distribution instead of the variance only, thus providing complementary insight on the inputs / output …
LTCI, Télécom Paris, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎电信学院，LTCI实验室）
Operator-Valued Kernels (OVKs) and associated vector-valued Reproducing Kernel Hilbert Spaces provide an elegant way to extend scalar kernel methods when the output space is a Hilbert space. Although primarily used in finite dimension for problems like multi-task regression, the ability of this framework to deal with infinite dimensional output spaces unlocks many more applications, such as functional regression, structured output prediction, and structured data representation. However, these sophisticated schemes crucially rely on the kernel trick in the output space, so that most of previous…
Department of Electrical Engineering, Stanford University（斯坦福大学电气工程系）；Department of Statistics, University of California, Berkeley（加州大学伯克利分校统计系）；Department of Statistics, Stanford University（斯坦福大学统计系）；Google Research, Brain Team（谷歌研究院大脑团队）
For a certain scaling of the initialization of stochastic gradient descent (SGD), wide neural networks (NN) have been shown to be well approximated by reproducing kernel Hilbert space (RKHS) methods. Recent empirical work showed that, for some classification tasks, RKHS methods can replace NNs without a large loss in performance. On the other hand, two-layers NNs are known to encode richer smoothness classes than RKHS and we know of special examples for which SGD-trained NN provably outperform RKHS. This is true even in the wide network limit, for a different scaling of the initialization. How…
The RKHS bandit problem (also called kernelized multi-armed bandit problem) is an online optimization problem of non-linear functions with noisy feedback. Although the problem has been extensively studied, there are unsatisfactory results for some problems compared to the well-studied linear bandit case. Specifically, there is no general algorithm for the adversarial RKHS bandit problem. In addition, high computational complexity of existing algorithms hinders practical application. We address these issues by considering a novel amalgamation of approximation theory and the misspecified linear …
Consider the problem: given the data pair $(\mathbf{x}, \mathbf{y})$ drawn from a population with $f_*(x) = \mathbf{E}[\mathbf{y} | \mathbf{x} = x]$, specify a neural network model and run gradient flow on the weights over time until reaching any stationarity. How does $f_t$, the function computed by the neural network at time $t$, relate to $f_*$, in terms of approximation and representation? What are the provable benefits of the adaptive representation by neural networks compared to the pre-specified fixed basis representation in the classical nonparametric literature? We answer the above qu…
Mahidol University（玛希隆大学）；Idiap Research Institute（Idiap研究所）；University of Bern（伯尔尼大学）；Centre of Excellence in Mathematics（数学卓越中心）
We focus on kernel methods for set-valued inputs and their application to Bayesian set optimization, notably combinatorial optimization. We investigate two classes of set kernels that both rely on Reproducing Kernel Hilbert Space embeddings, namely the ``Double Sum'' (DS) kernels recently considered in Bayesian set optimization, and a class introduced here called ``Deep Embedding'' (DE) kernels that essentially consists in applying a radial kernel on Hilbert space on top of the canonical distance induced by another kernel such as a DS kernel. We establish in particular that while DS kernels ty…
In this paper, we specify what functions induce the bounded composition operators on a reproducing kernel Hilbert space (RKHS) associated with an analytic positive definite function defined on $\mathbf{R}^d$. We prove that only affine transforms can do so in a pretty large class of RKHS. Our result covers not only the Paley-Wiener space on the real line, studied in previous works, but also much more general RKHSs corresponding to analytic positive definite functions where existing methods do not work. Our method only relies on an intrinsic properties of the RKHSs, and we establish a connection…
Weighted log-rank tests are arguably the most widely used tests by practitioners for the two-sample problem in the context of right-censored data. Many approaches have been considered to make weighted log-rank tests more robust against a broader family of alternatives, among them, considering linear combinations of weighted log-rank tests, and taking the maximum among a finite collection of them. In this paper, we propose as test statistic the supremum of a collection of (potentially infinite) weight-indexed log-rank tests where the index space is the unit ball in a reproducing kernel Hilbert …
In this paper, we consider the nonparametric least square regression in a Reproducing Kernel Hilbert Space (RKHS). We propose a new randomized algorithm that has optimal generalization error bounds with respect to the square loss, closing a long-standing gap between upper and lower bounds. Moreover, we show that our algorithm has faster finite-time and asymptotic rates on problems where the Bayes risk with respect to the square loss is small. We state our results using standard tools from the theory of least square regression in RKHSs, namely, the decay of the eigenvalues of the associated int…
Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximation
Department of Computing and Mathematical Sciences, Caltech（加州理工学院计算与数学科学系）；The Alan Turing Institute（艾伦·图灵研究所）；Department of Mechanical Engineering, Clemson University（克莱姆森大学机械工程系）
Lucerne University of Applied Sciences and Arts（卢塞恩应用科学与艺术大学）
We introduce a novel boosting algorithm called `KTBoost' which combines kernel boosting and tree boosting. In each boosting iteration, the algorithm adds either a regression tree or reproducing kernel Hilbert space (RKHS) regression function to the ensemble of base learners. Intuitively, the idea is that discontinuous trees and continuous RKHS regression functions complement each other, and that this combination allows for better learning of functions that have parts with varying degrees of regularity such as discontinuities and smooth parts. We empirically show that KTBoost significantly outp…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Pennsylvania State University（宾夕法尼亚州立大学）；The Institute of Statistical Mathematics（统计数学研究所）；University College London（伦敦大学学院）
Reproducing kernel Hilbert spaces (RKHSs) play an important role in many statistics and machine learning applications ranging from support vector machines to Gaussian processes and kernel embeddings of distributions. Operators acting on such spaces are, for instance, required to embed conditional probability distributions in order to implement the kernel Bayes rule and build sequential data models. It was recently shown that transfer operators such as the Perron-Frobenius or Koopman operator can also be approximated in a similar fashion using covariance and cross-covariance operators and that …
We propose a new point of view for regularizing deep neural networks by using the norm of a reproducing kernel Hilbert space (RKHS). Even though this norm cannot be computed, it admits upper and lower approximations leading to various practical strategies. Specifically, this perspective (i) provides a common umbrella for many existing regularization principles, including spectral norm and gradient penalties, or adversarial training, (ii) leads to new effective regularization penalties, and (iii) suggests hybrid strategies combining lower and upper bounds to get better approximations of the RKH…
School of Computer Science & Informatics, Cardiff University（卡迪夫大学计算机科学与信息学院）
A novel kernel-based support vector machine (SVM) for graph classification is proposed. The SVM feature space mapping consists of a sequence of graph convolutional layers, which generates a vector space representation for each vertex, followed by a pooling layer which generates a reproducing kernel Hilbert space (RKHS) representation for the graph. The use of a RKHS offers the ability to implicitly operate in this space using a kernel function without the computational complexity of explicitly mapping into it. The proposed model is trained in a supervised end-to-end manner whereby the convolut…
We aim to optimize a black-box function $f:\mathcal{X} \mapsto \mathbb{R}$ under the assumption that $f$ is Hölder smooth and has bounded norm in the RKHS associated with a given kernel $K$. This problem is known to have an agnostic Gaussian Process (GP) bandit interpretation in which an appropriately constructed GP surrogate model with kernel $K$ is used to obtain an upper confidence bound (UCB) algorithm. In this paper, we propose a new algorithm (\texttt{LP-GP-UCB}) where the usual GP surrogate model is augmented with Local Polynomial (LP) estimators of the Hölder smooth function $f$ to con…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Google Research（谷歌研究院）；University of Freiburg（弗赖堡大学）
We propose kernel distributionally robust optimization (Kernel DRO) using insights from the robust optimization theory and functional analysis. Our method uses reproducing kernel Hilbert spaces (RKHS) to construct a wide range of convex ambiguity sets, which can be generalized to sets based on integral probability metrics and finite-order moment bounds. This perspective unifies multiple existing robust and stochastic optimization methods. We prove a theorem that generalizes the classical duality in the mathematical problem of moments. Enabled by this theorem, we reformulate the maximization wi…
A fundamental task in kernel methods is to pick nodes and weights, so as to approximate a given function from an RKHS by the weighted sum of kernel translates located at the nodes. This is the crux of kernel density estimation, kernel quadrature, or interpolation from discrete samples. Furthermore, RKHSs offer a convenient mathematical and computational framework. We introduce and analyse continuous volume sampling (VS), the continuous counterpart -- for choosing node locations -- of a discrete distribution introduced in (Deshpande & Vempala, 2006). Our contribution is theoretical: we prove al…
LTCI, Télécom Paris, Institut Polytechnique de Paris（巴黎理工学院巴黎电信学院LTCI实验室）
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 reconstruction error. Beyond a first extension of the autoencoding scheme to possibly infinite dimensional Hilbert spaces, KAE further allows to autoencode any kind of data by choosing X to be itself a RKHS. A theoretical analysis of the model is carried out, providing a general…
Adaptive Kernel Selection for Kernelized Diffusion Maps
CY Cergy Paris University（塞纳-圣但尼大学）；Institut Polytechnique de Paris（巴黎理工学院）；California Institute of Technology（加州理工学院）；The Alan Turing Institute（阿兰·图灵研究所）；Imperial College London（伦敦帝国学院）
In this work, we develop and study an empirical projection operator scheme for solving nonparametric regression problems. This scheme is based on an approximate projection of the regression function over a suitable reproducing kernel Hilbert space (RKHS). The RKHS considered in this paper are generated by the Mercer kernels given by the Legendre Christoffel-Darboux and convolution Sinc kernels. We provide error and convergence analysis of the proposed scheme under the assumption that the regression function belongs to some suitable functional spaces. We also consider the popular RKHS regulariz…
