Boundedness of composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions

The use of regional covariance descriptors to generate feature data represented by Symmetric Positive Definite (SPD) matrices from images or videos has become increasingly common in machine learning. However, SPD data itself does not constitute a vector space, and dictionary learning involves a large number of linear operations, so dictionary learning cannot be performed directly on SPD data. For this reason, a more common method is to map the SPD data to the Reproducing Kernel Hilbert Space (RKHS). The so-called kernel learning is to find the most suitable RKHS for specific tasks. RKHS can be uniquely generated by a kernel function. Therefore, RKHS learning can also be considered as kernel learning. In this article, there are two main contributions. The first contribution is to propose a framework which based on Kernel Learning and Riemannian Metric (KLRM). Usually the learnable kernel function framework is to learn some parameters in the kernel function. The second contribution is dictionary learning by applying KLRM to SPD data. The SPD data is transformed into the RKHS generated by KLRM, and RKHS after training provides the most suitable working space for dictionary learning. Under the proposed framework, we design a positive definite kernel function, which is defined by the Log-Euclidean metric. This function can be transformed into a corresponding Riemannian kernel. The experimental results provided in this paper is compared with other state-of-the-art algorithms for SPD data dictionary learning and show that the proposed algorithm achieves better results.

In this paper, we shall state a simple and general meaning for reproducing kernels. We would like to answer a general and essential question that: what are reproducing kernels? By considering this basic problem, we were able to obtain a general concept of the generalized delta function as a generalized reproducing kernel and, as a general reproducing kernel Hilbert space, we can consider all separable Hilbert spaces consisting of functions.

Important operator characteristics (such boundedness or compactness) for particular classes of operators on particular reproducing kernel Hilbert spaces may be impacted by the behaviour of the operators on the reproducing kernels. These results have been shown for Toeplitz operators on the Paley-Wiener space, a reproducing kernel Hilbert space over C. Furthermore, we show how the norm of such an operator has no relation to the supremum of the norms of the pictures of the normalization reproducing kernels of the space. As a result, if this supremum is finite, the operator is implicitly bounded. To further demonstrate that the operator norm is not the same as the supremum of the norms of the pictures of the real normalized reproducing kernels, another example is also provided. We also set out a necessary and sufficient condition for the operators’ compactness in terms of their limiting function on the reproducing kernels.

We present a general nonlinear Bayesian filter for high-dimensional state estimation using the theory of reproducing kernel Hilbert space (RKHS). By applying the kernel method and the representer theorem to perform linear quadratic estimation in a functional space, we derive a Bayesian recursive state estimator for a general nonlinear dynamical system in the original input space. Unlike existing nonlinear extensions of the Kalman filter where the system dynamics are assumed known, the state-space representation for the Functional Bayesian Filter (FBF) is completely learned online from measurement data in the form of an infinite impulse response (IIR) filter or recurrent network in the RKHS, with universal approximation property. Using a positive definite kernel function satisfying Mercer’s conditions to compute and evolve information quantities, the FBF exploits both the statistical and time-domain information about the signal, extracts higher-order moments, and preserves the properties of covariances without the ill effects due to conventional arithmetic operations. We apply this novel kernel adaptive filtering (KAF) to recurrent network training, chaotic time-series estimation and cooperative filtering using Gaussian and non-Gaussian noises, and inverse kinematics modeling. Simulation results show FBF outperforms existing Kalman-based algorithms.

In this paper, we investigate the boundedness of composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We prove that the boundedness of a composition operator strongly limits the behavior of the original map, and it provides an effective method to investigate the properties of composition operators using the theory of dynamical system. Consequently, we prove that only affine maps can induce bounded composition operators on any quasi-Banach space continuously included in the space of entire functions of one variable if the function space contains a nonconstant function. We also prove that any polynomial automorphisms except affine transforms cannot induce bounded composition operators on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.

Isometric weighted composition operators on weighted Bergman spaces

Integration of feature vector selection and support vector machine for classification of imbalanced data

Many learning problems in computer vision can be posed as structured prediction problems, where the input and output instances are structured objects such as trees, graphs or strings rather than, single labels {+1, -1} or scalars. Kernel methods such as Structured Support Vector Machines, Twin Gaussian Processes (TGP), Structured Gaussian Processes, and vector-valued Reproducing Kernel Hilbert Spaces (RKHS), offer powerful ways to perform learning and inference over these domains. Positive definite kernel functions allow us to quantitatively capture similarity between a pair of instances over these arbitrary domains. A poor choice of the kernel function, which decides the RKHS feature space, often results in poor performance. Automatic kernel selection methods have been developed, but have focused only on kernels on the input domain (i.e.'one-way'). In this work, we propose a novel and efficient algorithm for learning kernel functions simultaneously, on both input and output domains. We introduce the idea of learning polynomial kernel transformations, and call this method Simultaneous Twin Kernel Learning (STKL). STKL can learn arbitrary, but continuous kernel functions, including 'one-way' kernel learning as a special case. We formulate this problem for learning covariances kernels of Twin Gaussian Processes. Our experimental evaluation using learned kernels on synthetic and several real-world datasets demonstrate consistent improvement in performance of TGP's.

The study of the hypercyclicity of an operator is an old problem in mathematics; it goes back to a paper of Birkhoff in 1929 proving the hypercyclicity of the translation operators in the space of all entire functions with the topology of uniform convergence on compact subsets. This article studies the hypercyclicity of translation operators in some general reproducing kernel Hilbert spaces of entire functions. These spaces are obtained by duality in a complex separable Hilbert space H by means of an analytic H-valued kernel. A link with the theory of de Branges spaces is also established. An illustrative example taken from the Hamburger moment problem theory is included.

Online pairwise learning algorithms with convex loss functions

We introduce a general nonparametric independence test between right-censored survival times and covariates, which may be multivariate. Our test statistic has a dual interpretation, first in terms of the supremum of a potentially infinite collection of weight-indexed log-rank tests, with weight functions belonging to a reproducing kernel Hilbert space (RKHS) of functions; and second, as the norm of the difference of embeddings of certain finite measures into the RKHS, similar to the Hilbert–Schmidt Independence Criterion (HSIC) test-statistic. We study the asymptotic properties of the test, finding sufficient conditions to ensure our test correctly rejects the null hypothesis under any alternative. The test statistic can be computed straightforwardly, and the rejection threshold is obtained via an asymptotically consistent Wild Bootstrap procedure. Extensive investigations on both simulated and real data suggest that our testing procedure generally performs better than competing approaches in detecting complex nonlinear dependence.

We consider the problem of conjugate phase retrieval in Paley-Wiener space $PW_{\pi}$. The goal of conjugate phase retrieval is to recover a signal $f$ from the magnitudes of linear measurements up to unknown phase factor and unknown conjugate, meaning $f(t)$ and $\overline{f(t)}$ are not necessarily distinguishable from the available data. We show that conjugate phase retrieval can be accomplished in $PW_{\pi}$ by sampling only on the real line by using structured convolutions. We also show that conjugate phase retrieval can be accomplished in $PW_{\pi}$ by sampling both $f$ and $f^{\prime}$ only on the real line. Moreover, we demonstrate experimentally that the Gerchberg-Saxton method of alternating projections can accomplish the reconstruction from vectors that do conjugate phase retrieval in finite dimensional spaces. Finally, we show that generically, conjugate phase retrieval can be accomplished by sampling at three times the Nyquist rate, whereas phase retrieval requires sampling at four times the Nyquist rate.

Learning based on kernel machines is widely known as a powerful tool for various fields of information science such as pattern recognition and regression estimation. An appropriate model selection is required in order to obtain desirable learning results. In our previous work, we discussed a class of kernels forming a nested class of reproducing kernel Hilbert spaces with an invariant metric and proved that the kernel corresponding to the smallest reproducing kernel Hilbert space, including an unknown true function, gives the best model. In this paper, we relax the invariant metric condition and show that a similar result is obtained when a subspace with an invariant metric exists.

The questions of dense definiteness and boundedness of composition operators in L 2spaces are studied by means of inductive limits of operators.Methods based on projective systems of measure spaces and inductive limits of L 2 -spaces are developed.Illustrative examples are presented.

We characterize all weighted composition operators v C ψ vC_{\psi } that commute with a J μ J_{\mu } -symmetric weighted composition operator u C φ uC_{\varphi } on the reproducing kernel Hilbert space H γ H_{\gamma } of analytic functions on the unit disk D {\mathbb D} . It turns out these commuting operators v C ψ vC_{\psi } are necessarily J μ J_{\mu } -symmetric. Furthermore, we obtain characterization(s) for the commuting operator v C ψ vC_{\psi } to be self-adjoint, normal or unitary.