Operator-valued Kernels Research Articles

Learning system parameters from turing patterns.

The Turing mechanism describes the emergence of spatial patterns due to spontaneous symmetry breaking in reaction-diffusion processes and underlies many developmental processes. Identifying Turing mechanisms in biological systems defines a challenging problem. This paper introduces an approach to the prediction of Turing parameter values from observed Turing patterns. The parameter values correspond to a parametrized system of reaction-diffusion equations that generate Turing patterns as steady state. The Gierer-Meinhardt model with four parameters is chosen as a case study. A novel invariant pattern representation based on resistance distance histograms is employed, along with Wasserstein kernels, in order to cope with the highly variable arrangement of local pattern structure that depends on the initial conditions which are assumed to be unknown. This enables us to compute physically plausible distances between patterns, to compute clusters of patterns and, above all, model parameter prediction based on training data that can be generated by numerical model evaluation with random initial data: for small training sets, classical state-of-the-art methods including operator-valued kernels outperform neural networks that are applied to raw pattern data, whereas for large training sets the latter are more accurate. A prominent property of our approach is that only a single pattern is required as input data for model parameter predicion. Excellent predictions are obtained for single parameter values and reasonably accurate results for jointly predicting all four parameter values.

KLMS algorithm in Vector-Valued RKHS for online nonlinear MIMO systems identification

This paper proposes an extension of the KLMS algorithm to the Vector-Valued Reproducing Kernel Hilbert Space (VV-RKHS), enabling the use of the operator-valued kernel (OV-kernel) in the online identification of multi-input multi-output (MIMO) nonlinear systems. The yielded multivariate kernel model offers more design flexibility and involves fewer parameters than other vector-valued KLMS algorithms present in the literature. Conditions ensuring the convergence of the proposed OV-KLMS algorithm are given. Experiments on a multivariate chaotic attractor as well as numerical simulations are carried out and show the effectiveness of the proposed algorithm

Stein Variational Gradient Descent on Infinite-Dimensional Space and Applications to Statistical Inverse Problems

In this paper, we propose an infinite-dimensional version of the Stein variational gradient descent (iSVGD) method for solving Bayesian inverse problems. The method can generate approximate samples from posteriors efficiently. Based on the concepts of operator-valued kernels and vector-valued reproducing kernel Hilbert spaces, a rigorous definition is given for the infinite-dimensional objects, e.g., the Stein operator, which are proved to be the limit of finite-dimensional ones. Moreover, a more efficient iSVGD with preconditioning operators is constructed by generalizing the change of variables formula and introducing a regularity parameter. The proposed algorithms are applied to an inverse problem of the steady state Darcy flow equation. Numerical results confirm our theoretical findings and demonstrate the potential applications of the proposed approach in the posterior sampling of large-scale nonlinear statistical inverse problems.

Operator-valued positive definite kernels and differentiable universality

We present a characterization for a positive definite operator-valued kernel to be universal or [Formula: see text]-universal, and apply these characterizations to a family of operator-valued kernels that are shown to be well behaved. Later, we obtain a characterization for an operator-valued differentiable kernel to be [Formula: see text]-universal and [Formula: see text]-universal. In order to obtain such characterization and examples, we generalize some well-known results concerning the structure of differentiable kernels to the operator-valued context. On the examples is given an emphasis on the radial kernels on Euclidean spaces.

Free noncommutative hereditary kernels: Jordan decomposition, Arveson extension, kernel domination

We discuss (i) a quantized version of the Jordan decomposition theorem for a complex Borel measure on a compact Hausdorff space, namely, the more general problem of decomposing a general noncommutative kernel (a quantization of the standard notion of kernel function) as a linear combination of completely positive noncommutative kernels (a quantization of the standard notion of positive definite kernel). Other special cases of (i) include: the problem of decomposing a general operator-valued kernel function as a linear combination of positive kernels (not always possible), of decomposing a general bounded linear Hilbert-space operator as a linear combination of positive linear operators (always possible), of decomposing a completely bounded linear map from a C^* -algebra \mathcal{A} to an injective C^* -algebra \mathcal{L}(\mathcal{Y}) as a linear combination of completely positive maps from \mathcal{A} to \mathcal{L}(\mathcal{Y}) (always possible). We also discuss (ii) a noncommutative kernel generalization of the Arveson extension theorem (any completely positive map \phi from an operator system \mathbb{S} to an injective C^* -algebra \mathcal{L}(\mathcal{Y}) can be extended to a completely positive map \phi_e from a C^* -algebra containing \mathbb{S} to \mathcal{L}(\mathcal{Y})) , and (iii) a noncommutative kernel version of a Positivstellensatz (i.e., finding a certificate to explain why one kernel is positive at points where another given kernel is strictly positive).

Singular stochastic integral operators

In this paper we introduce Calder\'on-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove $L^p$-extrapolation results under a H\"ormander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the $A_2$-conjecture. The results are applied to obtain $p$-independence and weighted bounds for stochastic maximal $L^p$-regularity both in the complex and real interpolation scale. As a consequence we obtain several new regularity results for the stochastic heat equation on $\mathbb{R}^d$ and smooth and angular domains.

Multilinear operator-valued Calderón-Zygmund theory

We develop a general theory of multilinear singular integrals with operator-valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the R-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of multilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic-probabilistic Banach space-valued analysis. In the bilinear case, we obtain a T(1)-type theorem without any additional assumptions on the Banach spaces other than the necessary UMD. Higher degrees of multilinearity are tackled via a new formulation of the Rademacher maximal function (RMF) condition. In addition to the natural UMD lattice cases, our RMF condition covers suitable tuples of non-commutative Lp-spaces. We employ our operator-valued theory to obtain new multilinear, multi-parameter, operator-valued theorems in the natural setting of UMD spaces with property α.

Boundedness of Singular Integral Operators with Operator-Valued Kernels and Maximal Regularity of Sectorial Operators in Variable Lebesgue Spaces

This paper is devoted to the maximal regularity of sectorial operators in Lebesgue spaces Lp⋅ with a variable exponent. By extending the boundedness of singular integral operators in variable Lebesgue spaces from scalar type to abstract-valued type, the maximal Lp⋅−regularity of sectorial operators is established. This paper also investigates the trace of the maximal regularity space E01,p⋅I, together with the imbedding property of E01,p⋅I into the range-varying function space C−I,X1−1/p⋅,p⋅. Finally, a type of semilinear evolution equations with domain-varying nonlinearities is taken into account.

An operator-valued T1 theory for symmetric CZOs

We provide a natural BMO-criterion for the L2-boundedness of Calderón-Zygmund operators with operator-valued kernels satisfying a symmetric property. Our arguments involve both classical and quantum probability theory. In the appendix, we give a proof of the L2-boundedness of the commutators [Rj,b] whenever b belongs to the Bourgain's vector-valued BMO space, where Rj is the j-th Riesz transform. A common ingredient is the operator-valued Haar multiplier studied by Blasco and Pott.

Compact vectorial Toeplitz operators on the Segal-Bargmann space

We provide a sufficient condition for the compactness of a Toeplitz operator acting on the Segal-Bargmann space of vector-valued functions written in terms of an associated operator-valued kernel.

Quantum integral equations of Volterra type in terms of discrete-time normal martingale

In this paper, we aim to introduce a quantum linear stochastic Volterra integral equation of convolution type with operator-valued kernels in a nuclear topological algebra. We first establish the existence and uniqueness of the solutions and give the explicit expression of the solutions. Then we prove the continuity, continuous dependence on free terms and other properties of the solution.

Spatiotemporal Pattern Extraction by Spectral Analysis of Vector-Valued Observables

We present a data-driven framework for extracting complex spatiotemporal patterns generated by ergodic dynamical systems. Our approach, called vector-valued spectral analysis (VSA), is based on an eigendecomposition of a kernel integral operator acting on a Hilbert space of vector-valued observables of the system, taking values in a space of functions (scalar fields) on a spatial domain. This operator is constructed by combining aspects of the theory of operator-valued kernels for multitask machine learning with delay-coordinate maps of dynamical systems. In contrast to conventional eigendecomposition techniques, which decompose the input data into pairs of temporal and spatial modes with a separable, tensor product structure, the patterns recovered by VSA can be manifestly non-separable, requiring only a modest number of modes to represent signals with intermittency in both space and time. Moreover, the kernel construction naturally quotients out dynamical symmetries in the data and exhibits an asymptotic commutativity property with the Koopman evolution operator of the system, enabling decomposition of multiscale signals into dynamically intrinsic patterns. Application of VSA to the Kuramoto–Sivashinsky model demonstrates significant performance gains in efficient and meaningful decomposition over eigendecomposition techniques utilizing scalar-valued kernels.

An operator-valued kernel associated with a commuting tuple of Hilbert space operators

We associate a one parameter family of positive definite E-valued kernels Ka,T with any commuting d-tuple T of bounded linear operators on a Hilbert space H, where a is a multi-sequence of non-zero complex numbers and E is an auxiliary Hilbert space. If Ha,T denotes the reproducing kernel Hilbert space associated with Ka,T, then there exists an isometry Ua,T from Ha,T into H. It turns out that Ua,T is surjective if and only E is a cyclic subspace for T. We apply the above scheme to the commuting toral Cauchy dual d-tuple St and the constant multi-sequence at with value 1 (resp. commuting spherical Cauchy dual d-tuple Ss and the multi-sequence as,α:=(d+|α|−1)!(d−1)!α!,α∈Nd) with E being the joint kernel of S⁎ to ensure an analytic model for S under some natural assumptions. In particular, the strictly higher dimensional obstruction to the intertwining of Ua,St with St (resp. the intertwining of Ua,Ss with Ss) and the multiplication tuple Mz is characterized in terms of a kernel condition. These results can be considered as toral and spherical analogs of Shimorin's Theorem (the case of d=1) stating that any left-invertible analytic operator admits an analytic model.

On stability of linear periodic differential difference equations

The paper considers the equationwhere the operator-valued bounded functions aj and bj are 2π-periodic, and the operator-valued kernels m and n are 2π-periodic with respect to the first argument. The connection between the input-output stability of the equation and the invertibility of a family of operators acting on the space of periodic functions is investigated.

On the $\ell^s$-boundedness of a family of integral operators

In this paper we prove an \ell^s -boundedness result for integral operators with operator-valued kernels. The proofs are based on extrapolation techniques with weights due to Rubio de Francia. The results will be applied by the first and third author in a subsequent paper where a new approach to maximal L^p -regularity for parabolic problems with time-dependent generator is developed.

Operator-valued dyadic shifts and the T(1) theorem

In this paper we extend dyadic shifts and the dyadic representation theorem to an operator-valued setting: We first define operator-valued dyadic shifts and prove that they are bounded. We then extend the dyadic representation theorem, which states that every scalar-valued Calder\'on-Zygmund operator can be represented as a series of dyadic shifts and paraproducts averaged over randomized dyadic systems, to operator-valued Calder\'on-Zygmund operators. As a corollary, we obtain another proof of the operator-valued, global T(1) theorem. We work in the setting of integral operators that have R-bounded operator-valued kernels and act on functions taking values in UMD-spaces. The domain of the functions is the Euclidean space equipped with the Lebesgue measure. In addition, we give new proofs for the following known theorems: Boundedness of the dyadic (operator-valued) paraproduct, a variant of Pythagoras' theorem for (vector-valued) functions adapted to a sparse collection of dyadic cubes, and a decoupling inequality for (UMD-valued) martingale differences.

ВЫРОЖДЕННЫЕ УРАВНЕНИЯ ВОЛЬТЕРРА ТИПА СВЕРТКИ В БАНАХОВЫХ ПРОСТРАНСТВАХ И ИХ ПРИЛОЖЕНИЯ

The article is devoted to the problem of unique solvability of linear integral and integral-differential Volterra equations in Banach spaces with irreversible operator in the main part. Operator-valued kernel has a special form, K ( t , s ) = g ( t – s ) A , where g = g ( t ) is a numeric function, and A is a linear operator. Abstract equations of this kind are very typical for applications. For the study of such equations it is possible to use structural stack theory of two linear operators, which has been developed by Professor G.A. Sviridyuk and his students. Another peculiarity of the studied problems is multiple zero of function g = g ( t ) at the point t = 0. Fundamental operator-functions of considered integral and integral-differential operators in Banach spaces are constructed under the assumption of relative spectrally boundness of operator A with respect to degenerated main part of equations. On this basis, theorems of unique existence of solutions in the class of distributions with left-bounded support are proved. The dependence between the order of singularity of generalized solutions and multiplicity of zero of integral kernel at the initial point is ascertained. Also we have obtained conditions under which generalized solutions are equal to the classical solutions. Theorems formulated for abstract equations are applied to the study of significant initial boundary value problems arising in plasma physics and mathematical theory of elasticity

Operator-valued kernels for learning from functional response data

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 ...

On local solvability and blow-up of solutions of an abstract nonlinear Volterra integral equation

A theorem on noncontinuable solutions is proved for abstract Volterra integral equations with operator-valued kernels (continuous and polar). It is shown that if there is no global solvability, then the C-norm of the solution is unbounded but does not tend to infinity in general. An example of Volterra equations whose noncontinuable solutions are unbounded but not infinitely large is constructed. It is shown that the theorems on noncontinuable solutions of the Cauchy problem for abstract equations of the first and nth kind (with a linear leading part) are special cases of the theorems proved in this paper.

Operator-valued kernel-based vector autoregressive models for network inference

Reverse-engineering of high-dimensional dynamical systems from time-course data still remains a challenging and important problem in knowledge discovery. For this learning task, a number of approaches primarily based on sparse linear models or Granger causality concepts have been proposed in the literature. However, when a system exhibits nonlinear dynamics, there does not exist a systematic approach that takes into account the nature of the underlying system. In this work, we introduce a novel family of vector autoregressive models based on different operator-valued kernels to identify the dynamical system and retrieve the target network that characterizes the interactions of its components. Assuming a sparse underlying structure, a key challenge, also present in the linear case, is to control the model's sparsity. This is achieved through the joint learning of the structure of the kernel and the basis vectors. To solve this learning task, we propose an alternating optimization algorithm based on proximal gradient procedures that learns both the structure of the kernel and the basis vectors. Results on the DREAM3 competition gene regulatory benchmark networks of sizes 10 and 100 show the new model outperforms existing methods. Another application of the model on climate data identifies interesting and interpretable interactions between natural and human activity factors, thus confirming the ability of the learning scheme to retrieve dependencies between state-variables.