We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics ...literature; on the other, maximum mean discrepancies (MMD), that is, distances between embeddings of distributions to reproducing kernel Hilbert spaces (RKHS), as established in machine learning. In the case where the energy distance is computed with a semimetric of negative type, a positive definite kernel, termed distance kernel, may be defined such that the MMD corresponds exactly to the energy distance. Conversely, for any positive definite kernel, we can interpret the MMD as energy distance with respect to some negative-type semimetric. This equivalence readily extends to distance covariance using kernels on the product space. We determine the class of probability distributions for which the test statistics are consistent against all alternatives. Finally, we investigate the performance of the family of distance kernels in two-sample and independence tests: we show in particular that the energy distance most commonly employed in statistics is just one member of a parametric family of kernels, and that other choices from this family can yield more powerful tests.
Contraction theory for dynamical systems on Euclidean spaces is well-established. For contractive (resp. semicontractive) systems, the distance (resp. semidistance) between any two trajectories ...decreases exponentially fast. For partially contractive systems, each trajectory converges exponentially fast to an invariant subspace. In this article, we develop contraction theory on Hilbert spaces. First, we provide a novel integral condition for contractivity, and for time-invariant systems, we establish the existence of a unique globally exponentially stable equilibrium. Second, we introduce the notions of partial and semicontraction and we provide various sufficient conditions for time-varying and time-invariant systems. Finally, we apply the theory on a classic reaction-diffusion system.
Despite various physical applications, state-space identification for continuous-time dynamic systems with noisy sampled data is less popular than discrete-time systems owing to unmeasurable state ...derivatives. Consequently, its scope is limited to linear or preparameterized systems. In this study, we formulate the identification as a (nonparametric) kernel regression problem applicable to nonlinear dynamic systems, which facilitates the joint assessment of the data fitness and fidelity of dynamics, that is, the error between the state derivative and the vector field. This is realized by employing "smooth" kernels, which facilitate the embedding of the state derivative into the reproducing kernel Hilbert space. To demonstrate the feasibility of the formulation, we extend the representer theorem, resulting in a finite dimensional albeit nonconvex minimization problem. To achieve fast linear estimation despite the nonconvexity, we propose a practical algorithm, called state-dynamics alternate descent, which is basically an unconstrained iterated quadratic programming. Furthermore, the proposed method was evaluated using the example of a simple nonlinear pendulum.
These lectures concentrate on (nonlinear) stochastic partial differential equations (SPDE) of evolutionary type. There are three approaches to analyze SPDE: the "martingale measure approach", the ..."mild solution approach" and the "variational approach". The purpose of these notes is to give a concise and as self-contained as possible an introduction to the "variational approach". A large part of necessary background material is included in appendices.
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 propose
kernel transfer operators
, which 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. The proposed numerical methods to compute empirical estimates of these kernel transfer operators subsume existing data-driven methods for the approximation of transfer operators such as extended dynamic mode decomposition and its variants. One main benefit of the presented kernel-based approaches is that they can be applied to any domain where a similarity measure given by a kernel is available. Furthermore, we provide elementary results on eigendecompositions of finite-rank RKHS operators. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics as well as video and text data analysis.
In this paper, a projection-type approximation method is introduced for solving a variational inequality problem. The proposed method involves only one projection per iteration and the underline ...operator is pseudo-monotone and L-Lipschitz-continuous. The strong convergence result of the iterative sequence generated by the proposed method is established, under mild conditions, in real Hilbert spaces. Sound computational experiments comparing our newly proposed method with the existing state of the art on multiple realistic test problems are given.
In this paper, we present a mathematical and computational framework for comparing and matching distributions in reproducing kernel Hilbert spaces (RKHS). This framework, called optimal transport in ...RKHS, is a generalization of the optimal transport problem in input spaces to (potentially) infinite-dimensional feature spaces. We provide a computable formulation of Kantorovich's optimal transport in RKHS. In particular, we explore the case in which data distributions in RKHS are Gaussian, obtaining closed-form expressions of both the estimated Wasserstein distance and optimal transport map via kernel matrices. Based on these expressions, we generalize the Bures metric on covariance matrices to infinite-dimensional settings, providing a new metric between covariance operators. Moreover, we extend the correlation alignment problem to Hilbert spaces, giving a new strategy for matching distributions in RKHS. Empirically, we apply the derived formulas under the Gaussianity assumption to image classification and domain adaptation. In both tasks, our algorithms yield state-of-the-art performances, demonstrating the effectiveness and potential of our framework.
Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted
L
2
spaces on the dual ...group. It is known that for suitably chosen subadditive weights, every such space is a Banach algebra with respect to pointwise multiplication of functions. In this paper, we study RKHSs associated with subconvolutive functions on the dual group. Sufficient conditions are established for these spaces to be symmetric Banach
∗
-algebras with respect to pointwise multiplication and complex conjugation of functions (here referred to as RKHAs). In addition, we study aspects of the spectra and state spaces of RKHAs. Sufficient conditions are established for an RKHA on a compact abelian group
G
to have the same spectrum as the
C
∗
-algebra of continuous functions on
G
. We also consider one-parameter families of RKHSs associated with semigroups of self-adjoint Markov operators on
L
2
(
G
)
, and show that in this setting subconvolutivity is a necessary and sufficient condition for these spaces to have RKHA structure. Finally, we establish embedding relationships between RKHAs and a class of Fourier–Wermer algebras that includes spaces of dominating mixed smoothness used in high-dimensional function approximation.
This paper studies modified extragradient methods with inertial extrapolation step and self-adaptive step-sizes for solving equilibrium problems in real Hilbert spaces. Strong convergence results are ...obtained under the assumption that the bifunction is pseudomonotone and satisfies the Lipchitz-type condition. Our method of proof is of independent interest and different from the recent arguments used in related papers on strong convergence methods with inertial steps for equilibrium problems. Numerical implementations and comparisons are given to support the theoretical findings.
The problem of multiple kernel learning based on penalized empirical risk minimization is discussed. The complexity penalty is determined jointly by the empirical L₂ norms and the reproducing kernel ...Hilbert space (RKHS) norms induced by the kernels with a data-driven choice of regularization parameters. The main focus is on the case when the total number of kernels is large, but only a relatively small number of them is needed to represent the target function, so that the problem is sparse. The goal is to establish oracle inequalities for the excess risk of the resulting prediction rule showing that the method is adaptive both to the unknown design distribution and to the sparsity of the problem.
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