The Tanimoto kernel (Jaccard index) is a well known tool to describe the similarity between sets of binary attributes. It has been extended to the case when the attributes are nonnegative real values. This paper introduces a more general Tanimoto kernel formulation which allows to measure the similarity of arbitrary real-valued functions. This extension is constructed by unifying the representation of the attributes via properly chosen sets. After deriving the general form of the kernel, explicit feature representation is extracted from the kernel function, and a simply way of including general kernels into the Tanimoto kernel is shown. Finally, the kernel is also expressed as a quotient of piecewise linear functions, and a smooth approximation is provided.
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE（法国国家信息与自动化研究所）
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 functional data analysis methods to nonlinear contexts. We study this problem in the case of kernel ridge regression for functional responses with an lr-norm constraint on the combination coefficients. The resulting optimization problem is more involved than those of multiple scalar-…
We consider the problem of learning a vector-valued function f in an online learning setting. The function f is assumed to lie in a reproducing Hilbert space of operator-valued kernels. We describe two online algorithms for learning f while taking into account the output structure. A first contribution is an algorithm, ONORMA, that extends the standard kernel-based online learning algorithm NORMA from scalar-valued to operator-valued setting. We report a cumulative error bound that holds both for classification and regression. We then define a second algorithm, MONORMA, which addresses the lim…
University of Seville（塞维利亚大学）；University Carlos III de Madrid（马德里卡洛斯三世大学）
Usually, complex-valued RKHS are presented as an straightforward application of the real-valued case. In this paper we prove that this procedure yields a limited solution for regression. We show that another kernel, here denoted as pseudo kernel, is needed to learn any function in complex-valued fields. Accordingly, we derive a novel RKHS to include it, the widely RKHS (WRKHS). When the pseudo-kernel cancels, WRKHS reduces to complex-valued RKHS of previous approaches. We address the kernel and pseudo-kernel design, paying attention to the kernel and the pseudo-kernel being complex-valued. In …
IBISC；Université d’Évry val d’Essonne（埃松省埃夫里大学）；LTCI；Télécom ParisTech（巴黎电信学院）；Université Paris-Saclay（巴黎萨克雷大学）；Department of Information and Computer Science（信息与计算机科学系）；Aalto University（阿尔托大学）
Devoted to multi-task learning and structured output learning, operator-valued kernels provide a flexible tool to build vector-valued functions in the context of Reproducing Kernel Hilbert Spaces. To scale up these methods, we extend the celebrated Random Fourier Feature methodology to get an approximation of operator-valued kernels. We propose a general principle for Operator-valued Random Fourier Feature construction relying on a generalization of Bochner's theorem for translation-invariant operator-valued Mercer kernels. We prove the uniform convergence of the kernel approximation for bound…
IBM Research（IBM研究院）；Istituto Italiano di Tecnologia（意大利理工学院）
We propose a general matrix-valued multiple kernel learning framework for high-dimensional nonlinear multivariate regression problems. This framework allows a broad class of mixed norm regularizers, including those that induce sparsity, to be imposed on a dictionary of vector-valued Reproducing Kernel Hilbert Spaces. We develop a highly scalable and eigendecomposition-free algorithm that orchestrates two inexact solvers for simultaneously learning both the input and output components of separable matrix-valued kernels. As a key application enabled by our framework, we show how high-dimensional…
IBM Research（IBM研究院）；Italian Institute of Technology（意大利理工学院）
We propose a general matrix-valued multiple kernel learning framework for high-dimensional nonlinear multivariate regression problems. This framework allows a broad class of mixed norm regularizers, including those that induce sparsity, to be imposed on a dictionary of vector-valued Reproducing Kernel Hilbert Spaces. We develop a highly scalable and eigendecomposition-free algorithm that orchestrates two inexact solvers for simultaneously learning both the input and output components of separable matrix-valued kernels. As a key application enabled by our framework, we show how high-dimensional…
Stein variational gradient descent (SVGD) is a particle-based inference algorithm that leverages gradient information for efficient approximate inference. In this work, we enhance SVGD by leveraging preconditioning matrices, such as the Hessian and Fisher information matrix, to incorporate geometric information into SVGD updates. We achieve this by presenting a generalization of SVGD that replaces the scalar-valued kernels in vanilla SVGD with more general matrix-valued kernels. This yields a significant extension of SVGD, and more importantly, allows us to flexibly incorporate various precond…
INRIA Lille - Nord Europe（里尔-北欧法国国家信息与自动化研究所）；Université de Bordeaux（波尔多大学）；Université de Lille（里尔大学）；Ecole Centrale de Lille（里尔中央理工学院）；Université de Rouen（鲁昂大学）
Although operator-valued kernels have recently received increasing interest in various machine learning and functional data analysis problems such as multi-task learning or functional regression, little attention has been paid to the understanding of their associated feature spaces. In this paper, we explore the potential of adopting an operator-valued kernel feature space perspective for the analysis of functional data. We then extend the Regularized Least Squares Classification (RLSC) algorithm to cover situations where there are multiple functions per observation. Experiments on a sound rec…
Aix-Marseille Université（艾克斯-马赛大学）；Ecole Centrale de Lille（里尔中央理工学院）；Université de Lille（里尔大学）；INSA de Rouen（鲁昂国立应用科学学院）；Université de Rouen（鲁昂大学）；ENS Cachan（卡尚高等师范学院）
In this paper we consider the problems of supervised classification and regression in the case where attributes and labels are functions: a data is represented by a set of functions, and the label is also a function. We focus on the use of reproducing kernel Hilbert space theory to learn from such functional data. Basic concepts and properties of kernel-based learning are extended to include the estimation of function-valued functions. In this setting, the representer theorem is restated, a set of rigorously defined infinite-dimensional operator-valued kernels that can be valuably applied when…
Aix Marseille Univ（艾克斯-马赛大学）；Université de Toulon（土伦大学）；CNRS（法国国家科学研究中心）；LIS（信息科学实验室）
We consider the problem of metric learning for multi-view data and present a novel method for learning within-view as well as between-view metrics in vector-valued kernel spaces, as a way to capture multi-modal structure of the data. We formulate two convex optimization problems to jointly learn the metric and the classifier or regressor in kernel feature spaces. An iterative three-step multi-view metric learning algorithm is derived from the optimization problems. In order to scale the computation to large training sets, a block-wise Nystr{ö}m approximation of the multi-view kernel matrix is …
Modeling dynamical systems with ordinary differential equations implies a mechanistic view of the process underlying the dynamics. However in many cases, this knowledge is not available. To overcome this issue, we introduce a general framework for nonparametric ODE models using penalized regression in Reproducing Kernel Hilbert Spaces (RKHS) based on operator-valued kernels. Moreover, we extend the scope of gradient matching approaches to nonparametric ODE. A smooth estimate of the solution ODE is built to provide an approximation of the derivative of the ODE solution which is in turn used to …
We study the stability properties of nonlinear multi-task regression in reproducing Hilbert spaces with operator-valued kernels. Such kernels, a.k.a. multi-task kernels, are appropriate for learning prob- lems with nonscalar outputs like multi-task learning and structured out- put prediction. We show that multi-task kernel regression algorithms are uniformly stable in the general case of infinite-dimensional output spaces. We then derive under mild assumption on the kernel generaliza- tion bounds of such algorithms, and we show their consistency even with non Hilbert-Schmidt operator-valued ke…
Universidad de Sevilla（塞维利亚大学）；The University of Edinburgh（爱丁堡大学）；The Alan Turing Institute（艾伦·图灵研究所）
We propose a novel adaptive kernel based regression method for complex-valued signals: the generalized complex-valued kernel least-mean-square (gCKLMS). We borrow from the new results on widely linear reproducing kernel Hilbert space (WL-RKHS) for nonlinear regression and complex-valued signals, recently proposed by the authors. This paper shows that in the adaptive version of the kernel regression for complex-valued signals we need to include another kernel term, the so-called pseudo-kernel. This new solution is endowed with better representation capabilities in complex-valued fields, since i…
INRIA Paris（法国国家信息与自动化研究所巴黎中心）；ENS - Département d’Informatique de l’École Normale Supérieure（巴黎高等师范学院计算机科学系）；PSL Research University（巴黎文理研究大学）
Shape constraints such as positive semi-definiteness (PSD) for matrices or convexity for functions play a central role in many applications in machine learning and sciences, including metric learning, optimal transport, and economics. Yet, very few function models exist that enforce PSD-ness or convexity with good empirical performance and theoretical guarantees. In this paper, we introduce a kernel sum-of-squares model for functions that take values in the PSD cone, which extends kernel sums-of-squares models that were recently proposed to encode non-negative scalar functions. We provide a re…
University of Oxford（牛津大学）；University College London（伦敦大学学院）；University of Cambridge（剑桥大学）；Imperial College London（帝国理工学院）；St. Petersburg State University（圣彼得堡国立大学）
Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to deploy Gaussian processes in novel areas of science, a rapidly-growing line of research has focused on constructively extending these models to handle non-Euclidean domains, including Riemannian manifolds, such as spheres and tori. We propose techniques that generalize this class to model vector fields on Riemannian manifolds, which are important in a number of application areas …
This paper presents a framework for computing random operator-valued feature maps for operator-valued positive definite kernels. This is a generalization of the random Fourier features for scalar-valued kernels to the operator-valued case. Our general setting is that of operator-valued kernels corresponding to RKHS of functions with values in a Hilbert space. We show that in general, for a given kernel, there are potentially infinitely many random feature maps, which can be bounded or unbounded. Most importantly, given a kernel, we present a general, closed form formula for computing a corresp…
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所（HIIT））；Aalto University（阿尔托大学）；Aix-Marseille University（艾克斯-马赛大学）；CNRS（法国国家科学研究中心）
We consider the problem of operator-valued kernel learning and investigate the possibility of going beyond the well-known separable kernels. Borrowing tools and concepts from the field of quantum computing, such as partial trace and entanglement, we propose a new view on operator-valued kernels and define a general family of kernels that encompasses previously known operator-valued kernels, including separable and transformable kernels. Within this framework, we introduce another novel class of operator-valued kernels called entangled kernels that are not separable. We propose an efficient two…
This paper studies the construction of a refinement kernel for a given operator-valued reproducing kernel such that the vector-valued reproducing kernel Hilbert space of the refinement kernel contains that of the given one as a subspace. The study is motivated from the need of updating the current operator-valued reproducing kernel in multi-task learning when underfitting or overfitting occurs. Numerical simulations confirm that the established refinement kernel method is able to meet this need. Various characterizations are provided based on feature maps and vector-valued integral representat…
Spectral Truncation Kernels: Noncommutativity in $C^*$-algebraic Kernel Machines
NTT, Inc.（NTT公司）；Center for Advanced Intelligence Project, RIKEN（RIKEN高级智能项目）；Graduate School of Mathematical Sciences, The University of Tokyo（东京大学数学科学研究生院）；Graduate School of Information Science and Technology, The university of Osaka（大阪大学信息科学与技术研究生院）；Department of Computer Science, Aix-Marseille University, CNRS, LIS（阿维尼昂-马赛大学计算机科学系，CNRS，LIS）
The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations
Fariborz Maseeh Department of Mathematics and Statistics（法里博兹·马塞厄数学与统计学系）；Portland State University（波特兰州立大学）；Translational Genomics Research Institute（转化基因组研究研究所）
NTT Network Technology Laboratories, NTT Corporation（NTT 公司网络技术实验室）；Graduate School of Science and Technology, Keio University（庆应义塾大学科学与技术研究生院）；Faculty of Science, Ehime University（爱媛大学理学部）；Center for Advanced Intelligence Project, RIKEN（理化学研究所革新智能综合研究中心）；Faculty of Science and Technology, Keio University（庆应义塾大学理工学部）；Institute of Mathematics for Industry, Kyushu University（九州大学产业数学研究所）
Kernel mean embedding (KME) is a powerful tool to analyze probability measures for data, where the measures are conventionally embedded into a reproducing kernel Hilbert space (RKHS). In this paper, we generalize KME to that of von Neumann-algebra-valued measures into reproducing kernel Hilbert modules (RKHMs), which provides an inner product and distance between von Neumann-algebra-valued measures. Von Neumann-algebra-valued measures can, for example, encode relations between arbitrary pairs of variables in a multivariate distribution or positive operator-valued measures for quantum mechanics…
Geneva School of Business Administration, HES-SO University of Applied Sciences of Western Switzerland（日内瓦商学院，瑞士西部应用科学与艺术大学）；University of Geneva（日内瓦大学）
Traditional linear methods for forecasting multivariate time series are not able to satisfactorily model the non-linear dependencies that may exist in non-Gaussian series. We build on the theory of learning vector-valued functions in the reproducing kernel Hilbert space and develop a method for learning prediction functions that accommodate such non-linearities. The method not only learns the predictive function but also the matrix-valued kernel underlying the function search space directly from the data. Our approach is based on learning multiple matrix-valued kernels, each of those composed …
Complex-valued signals are used in the modeling of many systems in engineering and science, hence being of fundamental interest. Often, random complex-valued signals are considered to be proper. A proper complex random variable or process is uncorrelated with its complex conjugate. This assumption is a good model of the underlying physics in many problems, and simplifies the computations. While linear processing and neural networks have been widely studied for these signals, the development of complex-valued nonlinear kernel approaches remains an open problem. In this paper we propose Gaussian…
We consider multi-objective optimization (MOO) of an unknown vector-valued function in the non-parametric Bayesian optimization (BO) setting, with the aim being to learn points on the Pareto front of the objectives. Most existing BO algorithms do not model the fact that the multiple objectives, or equivalently, tasks can share similarities, and even the few that do lack rigorous, finite-time regret guarantees that capture explicitly inter-task structure. In this work, we address this problem by modelling inter-task dependencies using a multi-task kernel and develop two novel BO algorithms base…
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…
Permutation-valued features arise in a variety of applications, either in a direct way when preferences are elicited over a collection of items, or an indirect way in which numerical ratings are converted to a ranking. To date, there has been relatively limited study of regression, classification, and testing problems based on permutation-valued features, as opposed to permutation-valued responses. This paper studies the use of reproducing kernel Hilbert space methods for learning from permutation-valued features. These methods embed the rankings into an implicitly defined function space, and …
Universidad de Sevilla（塞维利亚大学）；Swiss Data Science Center（瑞士数据科学中心）；University Carlos III de Madrid（马德里卡洛斯三世大学）；Stevens Institute of Technology（史蒂文斯理工学院）
In this paper we propose a novel Bayesian solution for nonlinear regression in complex fields. Previous solutions for kernels methods usually assume a complexification approach, where the real-valued kernel is replaced by a complex-valued one. This approach is limited. Based on results in complex-valued linear theory and Gaussian random processes we show that a pseudo-kernel must be included. This is the starting point to develop the new complex-valued formulation for Gaussian process for regression (CGPR). We face the design of the covariance and pseudo-covariance based on a convolution appro…
Nagoya University（名古屋大学）；RIKEN（理化学研究所）；Kyushu University（九州大学）
Understanding nonlinear dynamical systems (NLDSs) is challenging in a variety of engineering and scientific fields. Dynamic mode decomposition (DMD), which is a numerical algorithm for the spectral analysis of Koopman operators, has been attracting attention as a way of obtaining global modal descriptions of NLDSs without requiring explicit prior knowledge. However, since existing DMD algorithms are in principle formulated based on the concatenation of scalar observables, it is not directly applicable to data with dependent structures among observables, which take, for example, the form of a s…
Learning Operators by Regularized Stochastic Gradient Descent with Operator-valued Kernels
School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）；Fudan University（复旦大学）；Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（复旦大学当代应用数学重点实验室）
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
The necessary and sufficient conditions for existence of a generalized representer theorem are presented for learning Hilbert space-valued functions. Representer theorems involving explicit basis functions and Reproducing Kernels are a common occurrence in various machine learning algorithms like generalized least squares, support vector machines, Gaussian process regression and kernel based deep neural networks to name a few. Due to the more general structure of the underlying variational problems, the theory is also relevant to other application areas like optimal control, signal processing …
A Behavioral Framework for Data-Driven Modeling of Nonlinear Systems in Vector-Valued Reproducing Kernel Hilbert Spaces
Carl R. Woese Institute for Genomic Biology and the Coordinated Science Laboratory, University of Illinois Urbana-Champaign（卡尔·R·沃塞基因组生物学研究所和协调科学实验室，伊利诺伊大学厄巴纳-香槟分校）；Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois Urbana-Champaign（电气与计算机工程系和协调科学实验室，伊利诺伊大学厄巴纳-香槟分校）
