The goal of nonparametric regression is to recover an underlying regression function from noisy observations, under the assumption that the regression function belongs to a pre-specified infinite dimensional function space. In the online setting, when the observations come in a stream, it is generally computationally infeasible to refit the whole model repeatedly. There are as of yet no methods that are both computationally efficient and statistically rate-optimal. In this paper, we propose an estimator for online nonparametric regression. Notably, our estimator is an empirical risk minimizer (ERM) in a deterministic linear space, which is quite different from existing methods using random features and functional stochastic gradient. Our theoretical analysis shows that this estimator obtains rate-optimal generalization error when the regression function is known to live in a reproducing kernel Hilbert space. We also show, theoretically and empirically, that the computational expense of our estimator is much lower than other rate-optimal estimators proposed for this online setting.
In this article, we consider convergence rates in functional linear regression with functional responses, where the linear coefficient lies in a reproducing kernel Hilbert space (RKHS). Without assuming that the reproducing kernel and the covariate covariance kernel are aligned, or assuming polynomial rate of decay of the eigenvalues of the covariance kernel, convergence rates in prediction risk are established. The corresponding lower bound in rates is derived by reducing to the scalar response case. Simulation studies and two benchmark datasets are used to illustrate that the proposed approa…
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…
Regularized kernel methods such as support vector machines (SVM) and support vector regression (SVR) constitute a broad and flexible class of methods which are theoretically well investigated and commonly used in nonparametric classification and regression problems. As these methods are based on a Tikhonov regularization which is also common in inverse problems, this article investigates the use of regularized kernel methods for inverse problems in a unifying way. Regularized kernel methods are based on the use of reproducing kernel Hilbert spaces (RKHS) which lead to very good computational p…
We study the estimation and prediction of functional autoregressive~(FAR) processes, a statistical tool for modeling functional time series data. Due to the infinite-dimensional nature of FAR processes, the existing literature addresses its inference via dimension reduction and theoretical results therein require the (unrealistic) assumption of fully observed functional time series. We propose an alternative inference framework based on Reproducing Kernel Hilbert Spaces~(RKHS). Specifically, a nuclear norm regularization method is proposed for estimating the transition operators of the FAR pro…
Expectile, first introduced by Newey and Powell (1987) in the econometrics literature, has recently become increasingly popular in risk management and capital allocation for financial institutions due to its desirable properties such as coherence and elicitability. The current standard tool for expectile regression analysis is the multiple linear expectile regression proposed by Newey and Powell in 1987. The growing applications of expectile regression motivate us to develop a much more flexible nonparametric multiple expectile regression in a reproducing kernel Hilbert space. The resulting es…
We encounter a bottleneck when we try to borrow the strength of classical classifiers to classify functional data. The major issue is that functional data are intrinsically infinite dimensional, thus classical classifiers cannot be applied directly or have poor performance due to the curse of dimensionality. To address this concern, we propose to project functional data onto one specific direction, and then a distance-weighted discrimination DWD classifier is built upon the projection score. The projection direction is identified through minimizing an empirical risk function that contains the …
In this paper we investigate the problem of estimating the regression function in models with correlated observations. The data is obtained from several experimental units each of them forms a time series. We propose a new estimator based on the inverse of the autocovariance matrix of the observations, assumed known and invertible. Using the properties of the Reproducing Kernel Hilbert spaces, we give the asymptotic expressions of its bias and its variance. In addition, we give a theoretical comparison, by calculating the IMSE, between this new estimator and the classical one proposed by Gasse…
Imputation and propensity score weighting are two popular techniques for handling missing data. We address these problems using the regularized M-estimation techniques in the reproducing kernel Hilbert space. Specifically, we first use the kernel ridge regression to develop imputation for handling item nonresponse. While this nonparametric approach is potentially promising for imputation, its statistical properties are not investigated in the literature. Under some conditions on the order of the tuning parameter, we first establish the root-$n$ consistency of the kernel ridge regression imputa…
Non-linear systems of differential equations have attracted the interest in fields like system biology, ecology or biochemistry, due to their flexibility and their ability to describe dynamical systems. Despite the importance of such models in many branches of science they have not been the focus of systematic statistical analysis until recently. In this work we propose a general approach to estimate the parameters of systems of differential equations measured with noise. Our methodology is based on the maximization of the penalized likelihood where the system of differential equations is used…
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.
We generalize the orthonormal basis for the Gaussian RKHS described in \cite{MinhGaussian2010} to an infinite, continuously parametrized, family of orthonormal bases, along with some implications. The proofs are direct generalizations of those in \cite{MinhGaussian2010}.
An extension of reproducing kernel Hilbert space (RKHS) theory provides a new framework for modeling functional regression models with functional responses. The approach only presumes a general nonlinear regression structure as opposed to previously studied linear regression models. Generalized cross-validation (GCV) is proposed for automatic smoothing parameter estimation. The new RKHS estimate is applied to both simulated and real data as illustrations.
We review definitions and properties of reproducing kernel Hilbert spaces attached to Gaussian variables and processes, with a view to applications in nonparametric Bayesian statistics using Gaussian priors. The rate of contraction of posterior distributions based on Gaussian priors can be described through a concentration function that is expressed in the reproducing Hilbert space. Absolute continuity of Gaussian measures and concentration inequalities play an important role in understanding and deriving this result. Series expansions of Gaussian variables and transformations of their reprodu…
This paper addresses the problem of regression to reconstruct functions, which are observed with superimposed errors at random locations. We address the problem in reproducing kernel Hilbert spaces. It is demonstrated that the estimator, which is often derived by employing Gaussian random fields, converges in the mean norm of the reproducing kernel Hilbert space to the conditional expectation and this implies local and uniform convergence of this function estimator. By preselecting the kernel, the problem does not suffer from the curse of dimensionality. The paper analyzes the statistical prop…
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 …
This paper develops a frequentist solution to the functional calibration problem, where the value of a calibration parameter in a computer model is allowed to vary with the value of control variables in the physical system. The need of functional calibration is motivated by engineering applications where using a constant calibration parameter results in a significant mismatch between outputs from the computer model and the physical experiment. Reproducing kernel Hilbert spaces (RKHS) are used to model the optimal calibration function, defined as the functional relationship between the calibrat…
Kernel methods, being supported by a well-developed theory and coming with efficient algorithms, are among the most popular and successful machine learning techniques. From a mathematical point of view, these methods rest on the concept of kernels and function spaces generated by kernels, so called reproducing kernel Hilbert spaces. Motivated by recent developments of learning approaches in the context of interacting particle systems, we investigate kernel methods acting on data with many measurement variables. We show the rigorous mean field limit of kernels and provide a detailed analysis of…
We introduce kernel nonparametric tests for Lancaster three-variable interaction and for total independence, using embeddings of signed measures into a reproducing kernel Hilbert space. The resulting test statistics are straightforward to compute, and are used in powerful interaction tests, which are consistent against all alternatives for a large family of reproducing kernels. We show the Lancaster test to be sensitive to cases where two independent causes individually have weak influence on a third dependent variable, but their combined effect has a strong influence. This makes the Lancaster…
We show that polynomials do not belong to the reproducing kernel Hilbert space of infinitely differentiable translation-invariant kernels whose spectral measures have moments corresponding to a determinate moment problem. Our proof is based on relating this question to the problem of best linear estimation in continuous time one-parameter regression models with a stationary error process defined by the kernel. In particular, we show that the existence of a sequence of estimators with variances converging to $0$ implies that the regression function cannot be an element of the reproducing kernel…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Institute for Statistical Mathematics（统计数理研究所）；ETH Zürich（苏黎世联邦理工学院）
We describe a method to perform functional operations on probability distributions of random variables. The method uses reproducing kernel Hilbert space representations of probability distributions, and it is applicable to all operations which can be applied to points drawn from the respective distributions. We refer to our approach as {\em kernel probabilistic programming}. We illustrate it on synthetic data, and show how it can be used for nonparametric structural equation models, with an application to causal inference.
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 …
School of Informatics（信息学院）；University of Edinburgh（爱丁堡大学）；FU Berlin（柏林自由大学）
We present an embedding of stochastic optimal control problems, of the so called path integral form, into reproducing kernel Hilbert spaces. Using consistent, sample based estimates of the embedding leads to a model free, non-parametric approach for calculation of an approximate solution to the control problem. This formulation admits a decomposition of the problem into an invariant and task dependent component. Consequently, we make much more efficient use of the sample data compared to previous sample based approaches in this domain, e.g., by allowing sample re-use across tasks. Numerical ex…
In this paper, we introduce a new distribution regression model for probability distributions. This model is based on a Reproducing Kernel Hilbert Space (RKHS) regression framework, where universal kernels are built using Wasserstein distances for distributions belonging to W 2 ($Ω$) and $Ω$ is a compact subspace of R. We prove the universal kernel property of such kernels and use this setting to perform regressions on functions. Different regression models are first compared with the proposed one on simulated functional data. We then apply our regression model to transient evoked otoascoutic …
We introduce a novel data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space where balanced truncation may be carried out implicitly. This leads to a nonlinear reduction map which can be combined with a representation of the system belonging to a reproducing kernel Hilbert space to give a closed, reduced order dynamical system which captures the essential i…
Kernel based methods have shown effective performance in many remote sensing classification tasks. However their performance significantly depend on its hyper-parameters. The conventional technique to estimate the parameter comes with high computational complexity. Thus, the objective of this letter is to propose an fast and efficient method to select the bandwidth parameter of the Gaussian kernel in the kernel based classification methods. The proposed method is developed based on the operators in the reproducing kernel Hilbert space and it is evaluated on Support vector machines and PerTurbo…
The role of kernels is central to machine learning. Motivated by the importance of power-law distributions in statistical modeling, in this paper, we propose the notion of power-law kernels to investigate power-laws in learning problem. We propose two power-law kernels by generalizing Gaussian and Laplacian kernels. This generalization is based on distributions, arising out of maximization of a generalized information measure known as nonextensive entropy that is very well studied in statistical mechanics. We prove that the proposed kernels are positive definite, and provide some insights rega…
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
In this work, we present formulations for regularized Kullback-Leibler and Rényi divergences via the Alpha Log-Determinant (Log-Det) divergences between positive Hilbert-Schmidt operators on Hilbert spaces in two different settings, namely (i) covariance operators and Gaussian measures defined on reproducing kernel Hilbert spaces (RKHS); and (ii) Gaussian processes with squared integrable sample paths. For characteristic kernels, the first setting leads to divergences between arbitrary Borel probability measures on a complete, separable metric space. We show that the Alpha Log-Det divergences …
We propose a novel adaptive learning algorithm based on iterative orthogonal projections in the Cartesian product of multiple reproducing kernel Hilbert spaces (RKHSs). The task is estimating/tracking nonlinear functions which are supposed to contain multiple components such as (i) linear and nonlinear components, (ii) high- and low- frequency components etc. In this case, the use of multiple RKHSs permits a compact representation of multicomponent functions. The proposed algorithm is where two different methods of the author meet: multikernel adaptive filtering and the algorithm of hyperplane…
McGill University（麦吉尔大学）；Mila - Quebec Artificial Intelligence Institute（魁北克人工智能研究所（Mila））；Université du Québec à Montréal（魁北克大学蒙特利尔分校）；Facebook AI Research（Facebook人工智能研究）；CIFAR AI chair（CIFAR人工智能主席职位）；DeepMind；Université de Montréal（蒙特利尔大学）
We propose a general framework for policy representation for reinforcement learning tasks. This framework involves finding a low-dimensional embedding of the policy on a reproducing kernel Hilbert space (RKHS). The usage of RKHS based methods allows us to derive strong theoretical guarantees on the expected return of the reconstructed policy. Such guarantees are typically lacking in black-box models, but are very desirable in tasks requiring stability. We conduct several experiments on classic RL domains. The results confirm that the policies can be robustly embedded in a low-dimensional space…
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 …
This paper presents a stochastic behavior analysis of a kernel-based stochastic restricted-gradient descent method. The restricted gradient gives a steepest ascent direction within the so-called dictionary subspace. The analysis provides the transient and steady state performance in the mean squared error criterion. It also includes stability conditions in the mean and mean-square sense. The present study is based on the analysis of the kernel normalized least mean square (KNLMS) algorithm initially proposed by Chen et al. Simulation results validate the analysis.
Optimal experimental design seeks to determine the most informative allocation of experiments to infer an unknown statistical quantity. In this work, we investigate the optimal design of experiments for {\em estimation of linear functionals in reproducing kernel Hilbert spaces (RKHSs)}. This problem has been extensively studied in the linear regression setting under an estimability condition, which allows estimating parameters without bias. We generalize this framework to RKHSs, and allow for the linear functional to be only approximately inferred, i.e., with a fixed bias. This scenario captur…
The recent developments of basis pursuit and compressed sensing seek to extract information from as few samples as possible. In such applications, since the number of samples is restricted, one should deploy the sampling points wisely. We are motivated to study the optimal distribution of finite sampling points. Formulation under the framework of optimal reconstruction yields a minimization problem. In the discrete case, we estimate the distance between the optimal subspace resulting from a general Karhunen-Loeve transform and the kernel space to obtain another algorithm that is computationall…
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…
Development of metrics for structural data-generating mechanisms is fundamental in machine learning and the related fields. In this paper, we give a general framework to construct metrics on random nonlinear dynamical systems, defined with the Perron-Frobenius operators in vector-valued reproducing kernel Hilbert spaces (vvRKHSs). We employ vvRKHSs to design mathematically manageable metrics and also to introduce operator-valued kernels, which enables us to handle randomness in systems. Our metric provides an extension of the existing metrics for deterministic systems, and gives a specificatio…
This is a tutorial and survey paper on kernels, kernel methods, and related fields. We start with reviewing the history of kernels in functional analysis and machine learning. Then, Mercer kernel, Hilbert and Banach spaces, Reproducing Kernel Hilbert Space (RKHS), Mercer's theorem and its proof, frequently used kernels, kernel construction from distance metric, important classes of kernels (including bounded, integrally positive definite, universal, stationary, and characteristic kernels), kernel centering and normalization, and eigenfunctions are explained in detail. Then, we introduce types …
Zalando Research（扎兰多研究院）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Transfer operators such as the Perron--Frobenius or Koopman operator play an important role in the global analysis of complex dynamical systems. The eigenfunctions of these operators can be used to detect metastable sets, to project the dynamics onto the dominant slow processes, or to separate superimposed signals. We extend transfer operator theory to reproducing kernel Hilbert spaces and show that these operators are related to Hilbert space representations of conditional distributions, known as conditional mean embeddings in the machine learning community. Moreover, numerical methods to com…
In supervised learning using kernel methods, we often encounter a large-scale finite-sum minimization over a reproducing kernel Hilbert space (RKHS). Large-scale finite-sum problems can be solved using efficient variants of Newton method, where the Hessian is approximated via sub-samples of data. In RKHS, however, the dependence of the penalty function to kernel makes standard sub-sampling approaches inapplicable, since the gram matrix is not readily available in a low-rank form. In this paper, we observe that for this class of problems, one can naturally use kernel approximation to speed up t…
The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares $\sum_j(Y_j - μ(t_j))^2 + λ\int_a^b [μ"(t)]^2 dt$, where the data are $t_j,Y_j$, $j=1,..., n$. The minimization is taken over an infinite-dimensional function space, the space of all functions with square integrable second derivatives. But the calculations can be carried out in a finite-dimensional space. The reduction from minimizing over an infinite dimensional space to minimizing over a finite dimensional space occurs for more general objective functions: the data may…
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…
This paper presents a general vector-valued reproducing kernel Hilbert spaces (RKHS) framework for the problem of learning an unknown functional dependency between a structured input space and a structured output space. Our formulation encompasses both Vector-valued Manifold Regularization and Co-regularized Multi-view Learning, providing in particular a unifying framework linking these two important learning approaches. In the case of the least square loss function, we provide a closed form solution, which is obtained by solving a system of linear equations. In the case of Support Vector Mach…
Motivated by the problem of nonparametric inference in high level digital image analysis, we introduce a general extrinsic approach for data analysis on Hilbert manifolds with a focus on means of probability distributions on such sample spaces. To perform inference on these means, we appeal to the concept of neighborhood hypotheses from functional data analysis and derive a one-sample test. We then consider analysis of shapes of contours lying in the plane. By embedding the corresponding sample space of such shapes, which is a Hilbert manifold, into a space of Hilbert-Schmidt operators, we can…
