Some recent developments in the theory of the Bergman kernel function: a survey

Let Ĝ be a connected complex Lie group and X = Ĝ/Ĥ a complex manifold homogeneous under a holomorphic Ĝ-action. In order to understand X, e.g. how it fits into a fine classification, details of its function theory, etc., one should use as much Lie theoretic information about Ĝ as is possible. In particular it is often useful to study the orbit structure of real subgroups of Ĝ. Such orbits are usually not complex sub-manifolds of X.

Part I: Transmutations, Integral Equations and Special Functions.- Some Recent Developments in the Transmutation Operator Approach.- Transmutation Operators and Their Applications.- Hankel Generalized Convolutions with the Associated Legendre Functions in the Kernel and Their Applications.- Second Type Neumann Series Related to Nicholson's and to Dixon-Ferrar Formula.- On Some Generalizations of the Properties of the Multidimensional Generalized Erdelyi-Kober Operators and Their Applications.- Alternative Approach to Miller-Paris Transformations and Their Extensions.- Transmutation Operators For Ordinary Dunkl-Darboux Operators.- Theorems on Restriction of Fourier-Bessel and Multidimensional Bessel Transforms to Spherical Surfaces.- Necessary Condition for the Existence of an Intertwining Operator and Classification of Transmutations on Its Basis.- Polynomial Quantization on Line Bundles.- Fourier-Bessel Transforms of Measures and Qualitative Properties of Solutions of Singular Differential Equations.- Inversion of Hyperbolic B-Potentials.- One-Dimensional and Multi-Dimensional Integral Transforms of Buschman-Erdelyi Type with Legendre Functions in Kernels.- Distributions, Non-smooth Manifolds, Transmutations and Boundary Value Problems.- Part II: Transmutations in ODEs, Direct and Inverse Problems.- On a Transformation Operator Approach in the Inverse Spectral Theory of Integral and Integro-Differential Operators.- Expansion in Terms of Appropriate Functions and Transmutations.- Transmutation Operators as a Solvability Concept of Abstract Singular Equations.- On the Bessel-Wright Operator and Transmutation with Applications.- On a Method of Solving Integral Equation of Carleman Type on the Pair of Segments.- Transmutation Operators Boundary Value Problems.- Solution of Inverse Problems for Differential Operators with Delay.- Part III: Transmutations for Partial and Fractional Differential Equations.- Transmutations of the Composed Erdelyi-Kober Fractional Operators and Their Applications.- Distributed Order Equations in Banach Spaces with Sectorial Operators.- Transformation Operators for Fractional Order Ordinary Differential Equations and Their applications.- Strong Solutions of Semilinear Equations with Lower Fractional Derivatives.- Mean Value Theorems and Properties of Solutions of Linear Differential Equations.- Transmutations for Multi-Term Fractional Operators.- Fractional Bessel Integrals and Derivatives on Semi-axes.- The Fractional Derivative Expansion Method in Nonlinear Dynamics of Structures: A Memorial Essay.- Boundary Value Problem with Integral Condition for the Mixed Type Equation with a Singular Coefficient.

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.

This work presents a contemporary treatment of Krein's entire operators with deficiency indices $(1,1)$ and de Branges' Hilbert spaces of entire functions. Each of these theories played a central role in the research of both renown mathematicians. Remarkably, entire operators and de Branges spaces are intimately connected and the interplay between them has had an impact in both spectral theory and the theory of functions. This work exhibits the interrelation between Krein's and de Branges' theories by means of a functional model and discusses recent developments, giving illustrations of the main objects and applications to the spectral theory of difference and differential operators.

Visible light communication (VLC) has emerged as a viable complement to traditional radio frequency (RF) based systems and as an enabler for high data rate communications for beyond-5G (B5G) indoor communication systems. In particular, the emergence of new B5G-based applications with quality of service (QoS) requirements and massive connectivity has recently led to research on the required service-levels and the development of improved physical (PHY) layer methods. As part of recent VLC standards development activities, the IEEE has formed the 802.11bb “Light Communications (LC) for Wireless Local Area Networking” standardization group. This paper investigates the network requirements of 5G indoor services such as virtual reality (VR) and high-definition (HD) video for residential environments using VLC. In this paper, we consider such typical VLC scenarios with additional impairments such as light-emitting diode (LED) nonlinearity and imperfect channel feedback, and propose hyperparameter-free mitigation techniques using Reproducing Kernel Hilbert Space (RKHS) methods. In this context, we also propose using a direct current biased optical orthogonal frequency division multiplexing (DCO-OFDM)-based adaptive VLC transmission method that uses precomputed bit error rate (BER) expressions for these RKHS-based detection methods and performs adaptive BER-based modulation-order switching. Simulations of channel impulse responses (CIRs) show that the adaptive transmission method provides significantly improved error rate performance, which makes it promising for high data rate VLC-based 5G indoor services.

The demand for 3D models that represent real-world objects such as structures and buildings has increased in recent years. It is becoming increasingly important that the reconstructions are not only visually convincing but also feature high geometric accuracy. This includes, for example, the fields of civil engineering, terrestrial surveying and archeology, where precise measurements are made in the models for documentation and analysis purposes. There are different approaches to create such a reconstruction. The photogrammetric method Structure from Motion and laser scanning are among the most widely used methods here, as they do not require a complicated setup and can be used for scenarios at small to large scale. Recent developments are enabling unmanned robotic systems, especially sensor mounted UAVs, to assist in the recording of areas which are otherwise difficult to observe. The demand for a high geometric accuracy, however, comes at the expense of high computational complexity of up to several days. Hence, especially real-time reconstructions are unfeasible, such that recording and reconstruction procedure must be executed consecutively. The resulting model quality, i.e. completeness and accuracy, is only assessable afterwards. Since it is often difficult or even impossible to improve these models with additional measurements afterwards, methods that ensure a reliable acquisition of sufficient data is required.
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\nIn this thesis we develop new methods and theory that address this problem for the mentioned sensor types. For both, a probabilistic description of the expected surface reconstruction error is maintained cost-efficiently as an estimate for the model quality during the recording procedure. For image sensors this is realized by incrementally constructing confidence ellipsoids that describe the information obtained from all views. With depth sensors the surface quality is described by the variance of a Gaussian process implicit surface regression fit to point cloud data using polyharmonic kernel functions. Sensor poses are then assessed by the information they add to the subsequent reconstruction up to a desired geometric accuracy using a formulation that is motivated from Optimal Experimental Design. This quantity is further used in an iterative next-best-view selection framework as a subproblem of a coverage path planning problem.
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\nThe general formulations presented in this thesis enables a wide range of applications, such as offline and online view planning or various autonomous robot systems under consideration of dynamic and geometric constraints. We present the first multi-view coverage path planning approach, specifically targeted at autonomous Structure from Motion data acquisition. Its correctness is validated in simulation using the physics simulator Gazebo. Furthermore, we lay a foundation for similar applications with depth sensors. All presented algorithms were developed with scalability in mind and show promising results regarding real-time usability.

In the machine learning based on kernel tricks, people often put one variable of a kernel function on the given samples to produce the basic functions of a solution space of learning problem. If the collection of the given samples deviates from the data distribution, the solution space spanned by these basic functions will also deviate from the real solution space of learning problem. In this paper a multikernel-like learning algorithm based on data probability distribution (MKDPD) is proposed, in which the parameters of a kernel function are locally adjusted according to the data probability distribution, and thus produces different kernel functions. These different kernel functions will generate different Reproducing Kernel Hilbert Spaces (RKHS). The direct sum of the subspaces of these RKHS constitutes the solution space of learning problem. Furthermore, based on the proposed MKDPD algorithm, a new algorithm for labeling new coming data is proposed, in which the basic functions are retrained according to the new coming data, while the coefficients of the retrained basic functions remained unchanged to label the new coming data. The experimental results presented in this paper show the effectiveness of the proposed algorithms.

Modeling data relation as a hierarchical structure has proven beneficial for many learning scenarios, and the hyperbolic space, with negative curvature, can encode such data hierarchy without distortion. Several recent studies also show that the representation power of the hyperbolic space can be further improved by endowing the kernel methods. Unfortunately, the known kernel methods, developed in hyperbolic space, are limited by the adaptation capacity or distortion issues. This paper addresses the issues through a novel embedding function. To this end, we propose a curvature-aware isometric embedding, which establishes an isometry from the Poincar\'e model to a special reproducing kernel Hilbert space (RKHS). Then we can further define a series of kernels on this RKHS, including several positive definite kernels and an indefinite kernel. Thorough experiments are conducted to demonstrate the superiority of our proposals over existing-known hyperbolic and Euclidean kernels in various learning tasks, e.g., graph learning and zero-shot learning.

We propose a data analysis method that combines the objectives of nonlinear principal component analysis and nonlinear discriminant analysis with the kernel method in a reproducing kernel Hilbert space. This method addresses nonlinear data analysis problems in high-dimensional spaces, specifically the reproducing kernel Hilbert space, through the use of the kernel trick. Our proposed method can be considered as a semi-supervised data analysis approach. We evaluate our proposed method using various kernel functions and datasets, both visually and quantitatively. The evaluation results demonstrate that our proposal outperforms kernel principal component analysis and generalized discriminant analysis in terms of classification performance. This indicates the advantages and originality of our proposed method. Furthermore, we analyze and discuss our findings based on the evaluation results, and highlight potential areas for further research and future work related to our proposal.

Wavelet SVM in Reproducing Kernel Hilbert Space for hyperspectral remote sensing image classification

This article proposes a novel approach to linear dimension reduction for regression using nonparametric estimation with positive-definite kernels or reproducing kernel Hilbert spaces (RKHSs). The purpose of the dimension reduction is to find such directions in the explanatory variables that explain the response sufficiently: this is called sufficient dimension reduction. The proposed method is based on an estimator for the gradient of the regression function considered for the feature vectors mapped into RKHSs. It is proved that the method is able to estimate the directions that achieve sufficient dimension reduction. In comparison with other existing methods, the proposed one has wide applicability without strong assumptions on the distributions or the type of variables, and needs only eigendecomposition for estimating the projection matrix. The theoretical analysis shows that the estimator is consistent with certain rate under some conditions. The experimental results demonstrate that the proposed method successfully finds effective directions with efficient computation even for high-dimensional explanatory variables.

Covariance matrices have attracted increasing attention for data representation in many computer vision tasks. The nonsingular covariance matrices are regarded as points on Riemannian manifolds rather than Euclidean space. A common technique for classification on Riemannian manifolds is to embed the covariance matrices into a reproducing kernel Hilbert space (RKHS), and then construct a map from RKHS to Euclidean space, while the explicit map from RKHS to Euclidean space in most kernel-based methods only depends on a linear hypothesis. In this paper, we propose a subspace learning framework to project Riemannian manifolds to Euclidean space, and give the theoretical derivation for it. Specifically, the Euclidean space is isomorphic to the subspace of RKHS. Under the framework, firstly we define an improved Log-Euclidean Gaussian radial basis function kernel for embedding. The first order statistical features of input images are incorporated into the kernel function to increase the discriminative power. After that we seek the optimal projection matrix of the subspace of the RKHS by conducting a graph embedding discriminant analysis. Texture recognition and object categorization experiments with region covariance descriptors demonstrate the considerable effectiveness of the improved Log-Euclidean Gaussian RBK kernel and the proposed method.

Various kernel-based methods have been developed with great success in many fields, but very little research has been published that is concerned with a multiple attribute kernel in reproducing kernel Hilbert space (RKHS). In this paper, we propose a novel elastic kernel called a multiple attribute convolution kernel with reproducing property (MACKRP) and present improved classification results over conventional approaches in the RKHS rather than the more commonly used Hilbert space. The MACKRP consists of two major steps. First, we find the basic solution of a generalized differential operator by the delta function, and then we design a convolution function using this solution. This convolution function is proven to be a specific reproducing kernel called a convolution reproducing kernel (CRK) in H3-space. Second, we prove that the CRK satisfies the condition of Mercer kernel. And the CRK is composed of three attributes (L1-norm, L2-norm and Laplace kernel), and each attribute can capture a different feature, with all attributes generating a novel kernel which we call an MACKRP. The experimental results demonstrate that the MACKRP possesses approximation and regularization performance and that classification results are consistently comparable or superior to a number of other state-of-the-art kernel functions.

Automatic target recognition has been widely studied over the years, yet it is still an open problem. The main obstacle consists in extended operating conditions, e.g.., depression angle change, configuration variation, articulation, and occlusion. To deal with them, this paper proposes a new classification strategy. We develop a new representation model via the steerable wavelet frames. The proposed representation model is entirely viewed as an element on Grassmann manifolds. To achieve target classification, we embed Grassmann manifolds into an implicit reproducing Kernel Hilbert space (RKHS), where the kernel sparse learning can be applied. Specifically, the mappings of training sample in RKHS are concatenated to form an overcomplete dictionary. It is then used to encode the counterpart of query as a linear combination of its atoms. By designed Grassmann kernel function, it is capable to obtain the sparse representation, from which the inference can be reached. The novelty of this paper comes from: 1) the development of representation model by the set of directional components of Riesz transform; 2) the quantitative measure of similarity for proposed representation model by Grassmann metric; and 3) the generation of global kernel function by Grassmann kernel. Extensive comparative studies are performed to demonstrate the advantage of proposed strategy.

The nonparametric multivariate analysis of variance (NPMANOVA) testing procedure has been proven to be a valuable tool for comparing groups. In the present paper, we propose a kernel extension of this technique in order to effectively confront high-dimensionality, a recurrent problem in many fields of science. The new method is called kernel multivariate analysis of variance (KMANOVA). The basic idea is to take advantage of the kernel framework: we propose to project the data from the original data space to a Hilbert space generated by a given kernel function and then perform the NPMANOVA method in the reproducing kernel Hilbert space (RKHS). Dispersion of the embedded points can be measured by the distance induced by the inner product in the RKHS but also by many other distances best suited in high-dimensional settings. For this purpose, we study two promising distances: a Manhattan-type distance and a distance based on an orthogonal projection of the embedded points in the direction of the group centroids. We show that the NPMANOVA method and the KMANOVA method with the induced distance are essentially equivalent. We also show that the KMANOVA method with the other two distances performs considerably better than the NPMANOVA method. We illustrate the advantages of our approach in the context of genetic association studies and demonstrate its usefulness on Alzheimer’s disease data. We also provide a software implementation of the method that is available on GitHub https://github.com/8699vicente/Kmanova.