Positive Semi-definite Function Research Articles

Rudin’s extension theorems and exponential convexity for matrix- and function-valued positive semidefinite functions

Rudin’s extension theorems and exponential convexity for matrix- and function-valued positive semidefinite functions

Extensions of k-regulous functions from two-dimensional varieties

AbstractWe prove that a k-regulous function defined on a two-dimensional non-singular affine variety can be extended to an ambient variety. Additionally we derive some results concerning sums of squares of k-regulous functions; in particular we show that every positive semi-definite regular function on a non-singular affine variety can be written as a sum of squares of locally Lipschitz regulous functions.

Convergence Analysis of Value Iteration Adaptive Dynamic Programming for Continuous-Time Nonlinear Systems.

This article is concerned with the convergence property and error bounds analysis of value iteration (VI) adaptive dynamic programming for continuous-time (CT) nonlinear systems. The size relationship between the total value function and the single integral step cost is described by assuming a contraction assumption. Then, the convergence property of VI is proved while the initial condition is an arbitrary positive semidefinite function. Moreover, the accumulated effects of approximation errors generated in each iteration are taken into consideration while using approximators to implement the algorithm. Based on the contraction assumption, the error bounds condition is proposed, which ensures the approximated iterative results converge to a neighborhood of the optimum, and the relation between the optimal solution and approximated iterative results is also derived. To make the contraction assumption more concrete, an estimation way is proposed to derive a conservative value of the assumption. Finally, three simulation cases are given to validate the theoretical results.

Stability and Performance Analysis of NMPC: Detectable Stage Costs and General Terminal Costs

We provide a stability and performance analysis for nonlinear model predictive control (NMPC) schemes subject to input constraints. Given an exponential stabilizability and detectability condition w.r.t. the employed state cost, we provide a sufficiently long prediction horizon to ensure asymptotic stability and a desired performance bound w.r.t. the infinite-horizon optimal controller. Compared to existing results, the provided analysis is applicable to positive semi-definite (detectable) cost functions, provides tight bounds using a linear programming analysis, and allows for a seamless integration of general positive-definite terminal cost functions in the analysis. The practical applicability of the derived theoretical results are demonstrated with numerical examples.

Extending the Gneiting class for modeling spatially isotropic and temporally symmetric vector random fields

We provide new classes of nonseparable univariate and multivariate space-time covariance functions that extend the Gneiting class. In particular, we prove that the spatial generator of the Gneiting class does not need to be completely monotone and can be replaced with the radial profile of a continuous positive semidefinite function in a finite dimensional Euclidean space, and we weaken the assumptions on the temporal margins so as to include more general behaviors than that of the original Gneiting construction. Our findings allow for covariance functions that are compactly-supported in space and/or nonmonotone in time, i.e., they offer versatility to model complex features commonly observed on data indexed with space-time coordinates.

Improved robust reduced-order sliding mode fault-tolerant control for T-S fuzzy systems with disturbances

This work investigates the problem of robust sliding mode fault-tolerant control for Takagi-Sugeno (T-S) fuzzy systems with disturbances. By virtue of a nonlinear sliding variable with an exponential decaying term, a reduced-order fuzzy sliding motion is established for the system throughout the entire responses. Since the existence of unforeseen actuator faults and disturbances affects the transient performance of sliding variable, which leads to poor performance of sliding motion system. To mitigate the influences of faults and disturbances on the sliding motion system, a class of positive semi-definite barrier functions is introduced to design controller, which ensures the sliding variable within a prescribed upper bound independent to actuator faults and disturbances. Finally, through the Lyapunov direct method, some sufficient conditions are derived such that the closed-loop system is uniformly bounded with time-vary performance boundary. In simulation part, several examples are given to demonstrate the effectiveness of the theoretical results.

Practical bipartite consensus for multi-agent systems: A barrier function-based adaptive sliding-mode control approach

This paper is concerned with bipartite consensus tracking for multi-agent systems with unknown disturbances. A barrier function-based adaptive sliding-mode control (SMC) approach is proposed such that the bipartite steady-state error is converged to a predefined region of zero in finite time. Specifically, based on an error auxiliary taking neighboring antagonistic interactions into account, an SMC law is designed with an adaptive gain. The gain can switch to a positive semi-definite barrier function to ensure that the error auxiliary is constrained to a predefined neighborhood of zero, which in turn guarantees practical bipartite consensus tracking. A distinguished feature of the proposed controller is its independence on the bound of disturbances, while the input chattering phenomenon is alleviated. Finally, a numerical example is provided to verify the effectiveness of the proposed controller.

Compactly-Supported Isotropic Covariances on Spheres Obtained from Matrix-Valued Covariances in Euclidean Spaces

Let p, d be positive integers, with d odd. Let \(\varvec{\phi }:[0,+\infty ) \rightarrow {\mathbb {R}}^{p \times p}\) be the isotropic part of a matrix-valued and isotropic covariance function (a positive semidefinite matrix-valued function) that is defined over the d-dimensional Euclidean space. If \(\varvec{\phi }\) is compactly supported over \([0,\pi ]\), then we show that the restriction of \(\varvec{\phi }\) to \([0,\pi ]\) is the isotropic part of a matrix-valued covariance function defined on a d-dimensional sphere, where isotropy in this case means that the covariance function depends on the geodesic distance. Our result does not need any assumption of continuity for the mapping \(\varvec{\phi }\). Further, when \(\varvec{\phi }\) is continuous, we provide an analytical expression of the d-Schoenberg sequence associated with the compactly-supported covariance on the sphere, which only requires knowledge of the Fourier transform of its isotropic part, and illustrate with the Gauss hypergeometric covariance model, which encompasses the well-known spherical, cubic, Askey and generalized Wendland covariances, and with a hole effect covariance model. Special cases of the results presented in this paper have been provided by other authors in the past decade.

An inverse problem for a class of lacunary canonical systems with diagonal Hamiltonian

Hamiltonians are 2-by-2 positive semidefinite real symmetric matrix-valued functions satisfying certain conditions. In this paper, we solve the inverse problem for which recovers a Hamiltonian from the solution of a first-order system attached to a given Hamiltonian, consisting of ordinary differential equations parametrized by a set of complex numbers, under certain conditions for the solutions. This inverse problem is a generalization of the inverse problem for two-dimensional canonical systems.

Entropy production in a longitudinally expanding Yang–Mills field with use of the Husimi function: semiclassical approximation

Abstract We investigate the possible thermalization process of the highly occupied and weakly coupled Yang–Mills fields expanding along the beam axis through an evaluation of the entropy, particle number, and pressure anisotropy. The time evolution of the system is calculated by solving the equation of motion for the Wigner function in the semiclassical approximation with initial conditions mimicking the glasma. For the evaluation of the entropy, we adopt Husimi–Wehrl (HW) entropy, which is obtained by using the Husimi function, a positive semidefinite quantum distribution function given by smearing the Wigner function. By numerical calculations at g = 0.1 and 0.2, the entropy production is found to occur together with the particle creation in two distinct stages: In the first stage, the particle number and entropy at low longitudinal momenta grow rapidly. In the second stage, the particle number and entropy of higher longitudinal momentum modes show a slower increase. The pressure anisotropy remains in our simulation and implies that the system is still out of equilibrium.

Criteria and characterizations for spatially isotropic and temporally symmetric matrix-valued covariance functions

We consider spatial matrix-valued isotropic covariance functions in Euclidean spaces and provide a very short proof of a celebrated characterization result proposed by earlier literature. We then provide a characterization theorem to create a bridge between a class of matrix-valued functions and the class of matrix-valued positive semidefinite functions in finite-dimensional Euclidean spaces. We culminate with criteria of the Pólya type for matrix-valued isotropic covariance functions, and with a generalization of Schlather’s class of multivariate spatial covariance functions. We then challenge the problem of matrix-valued space–time covariance functions, and provide a general class that encompasses all the proposals on the Gneiting nonseparable class provided by earlier literature.

Adaptive incremental sliding mode control for a robot manipulator

An adaptive incremental sliding mode control (AISMC) scheme for a robot manipulator is presented in this paper. Firstly, an incremental backstepping (IBS) controller is designed using time-delay estimation (TDE) to reduce dependence on the mathematical model. After substituting IBS controller into the nonlinear system, a linear system w.r.t. tracking errors is obtained while TDE error is the disturbance. Then, the AISMC scheme, including a nominal controller and an SMC, is developed for the resulted linear system to improve control performance. According to the equivalent control method, the SMC in the AISMC scheme is to handle TDE error. To receive optimal control performance at the sliding manifold, an LQR controller is selected as the nominal controller. The SMC is designed based on positive semi-definite barrier function (PSDBF) since it prevents switching gains from being over/under-estimated, and two practical problems are addressed in this paper: A new PSDBF is designed and conservative (large) setting bounds affecting tracking precision and/or system stability are avoided; An improved PSDBF based SMC is developed where the PSDBF and an adaptive parameter are used simultaneously to regulate switching gains, and the system is still stable when sliding variable occasionally exceeds the predefined vicinity. Moreover, finite-time convergence property of the sliding variable is strictly analyzed. Finally, real-time experiments are conducted to verify the effectiveness of the proposed control method.

Context-aware deep kernel networks for image annotation

Context plays a crucial role in visual recognition as it provides complementary clues for different learning tasks including image classification and annotation. As the performances of these tasks are currently reaching a plateau, any extra knowledge, including context, should be leveraged in ordficant leaps in these performances. In the particular scenario of kernel machines, context-aware kernel design aims at learning positive semi-definite similarity functions which return high values not only when data share similar contents, but also similar structures (a.k.a. contexts). However, the use of context in kernel design has not been fully explored; indeed, context in these solutions is handcrafted instead of being learned.In this paper, we introduce a novel deep network architecture that learns context in kernel design. This architecture is fully determined by the solution of an objective function mixing a content term that captures the intrinsic similarity between data, a context criterion which models their structure and a regularization term that helps designing smooth kernel network representations. The solution of this objective function defines a particular deep network architecture whose parameters correspond to different variants of learned contexts including layerwise, stationary and classwise; larger values of these parameters correspond to the most influencing contextual relationships between data. Extensive experiments conducted on the challenging ImageCLEF Photo Annotation, Corel5k and NUS-WIDE benchmarks show that our deep context networks are highly effective for image classification and the learned contexts further enhance the performance of image annotation.

Analytical continuation of matrix-valued functions: Carathéodory formalism

Finite-temperature quantum field theories are formulated in terms of Green's functions and self-energies on the Matsubara axis. In multi-orbital systems, these quantities are related to positive semidefinite matrix-valued functions of the Carath\'eodory and Schur class. Analysis, interpretation and evaluation of derived quantities such as real-frequency response functions requires analytic continuation of the off-diagonal elements to the real axis. We derive the criteria under which such functions exist for given Matsubara data and present an interpolation algorithm that intrinsically respects their mathematical properties. For small systems with precise Matsubara data, we find that the continuation exactly recovers all off-diagonal and diagonal elements. In real-materials systems, we show that the precision of the continuation is sufficient for the analytic continuation to commute with the Dyson equation, and we show that the commonly used truncation of off-diagonal self-energy elements leads to considerable approximation artifacts. Our method paves the way for the systematic evaluation of Matsubara data with equations of many-body theory on the real-frequency axis.

Characterizations of Herglotz\u2013Nevanlinna Functions Using Positive Semi-Definite Functions and the Nevanlinna Kernel in Several Variables

In this paper, we give several characterizations of Herglotz–Nevanlinna functions in terms of a specific type of positive semi-definite functions called Poisson-type functions. This allows us to propose a multidimensional analogue of the classical Nevanlinna kernel and a definition of generalized Nevanlinna functions in several variables. Furthermore, a characterization of the symmetric extension of a Herglotz–Nevanlinna function is also given. The subclass of Loewner functions is discussed as well, along with an interpretation of the main result in terms of holomorphic functions on the unit polydisk with non-negative real part.

Value Iteration-Based H∞ Controller Design for Continuous-Time Nonlinear Systems Subject to Input Constraints

In this paper, a novel integral reinforcement learning method is proposed based on value iteration (VI) to design the $H_{\infty }$ controller for continuous-time nonlinear systems subject to input constraints. To confront the control constraints, a nonquadratic function is introduced to reconstruct the ${L_{2}}$ -gain condition for the $H_{\infty }$ control problem. Then, the VI method is proposed to solve the corresponding Hamilton–Jacobi–Isaacs equation initialized with an arbitrary positive semi-definite value function. Compared with most existing works developed based on policy iteration, the initial admissible control policy is no longer required which results in a more free initial condition. The iterative process of the proposed VI method is analyzed and the convergence to the saddle point solution is proved in a general way. For the implementation of the proposed method, only one neural network is introduced to approximate the iterative value function, which results in a simpler architecture with less computational load compared with utilizing three neural networks. To verify the effectiveness of the VI-based method, two nonlinear cases are presented, respectively.

Learning Deep Graph Representations via Convolutional Neural Networks

Graph-structured data arise in many scenarios. A fundamental problem is to quantify the similarities of graphs for tasks such as classification. R-convolution graph kernels are positive-semidefinite functions that decompose graphs into substructures and compare them. One problem in the effective implementation of this idea is that the substructures are not independent, which leads to high-dimensional feature space. In addition, graph kernels cannot capture the high-order complex interactions between vertices. To mitigate these two problems, we propose a framework called DeepMap to learn deep representations for graph feature maps. The learned deep representation for a graph is a dense and low-dimensional vector that captures complex high-order interactions in a vertex neighborhood. DeepMap extends Convolutional Neural Networks (CNNs) to arbitrary graphs by generating aligned vertex sequences and building the receptive field for each vertex. We empirically validate DeepMap on various graph classification benchmarks and demonstrate that it achieves state-of-the-art performance.

Improved value iteration for neural-network-based stochastic optimal control design

In this paper, a novel value iteration adaptive dynamic programming (ADP) algorithm is presented, which is called an improved value iteration ADP algorithm, to obtain the optimal policy for discrete stochastic processes. In the improved value iteration ADP algorithm, for the first time we propose a new criteria to verify whether the obtained policy is stable or not for stochastic processes. By analyzing the convergence properties of the proposed algorithm, it is shown that the iterative value functions can converge to the optimum. In addition, our algorithm allows the initial value function to be an arbitrary positive semi-definite function. Finally, two simulation examples are presented to validate the effectiveness of the developed method.

Global Sobolev inequalities and degenerate p-Laplacian equations

We prove that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of p-Laplacian equations. Given Ω⊂Rn, let ρ be a quasi-metric on Ω, and let Q be an n×n positive semi-definite matrix function defined on Ω. For an open set Θ⋐Ω, we give sufficient conditions to show that if the local weak Sobolev inequality(⨏B|f|pσdx)1pσ≤C[r(B)⨏B|Q∇f|pdx+⨏B|f|pdx]1p holds for some σ>1, all balls B⊂Θ, and functions f∈Lip0(Θ), then the global Sobolev inequality(∫Θ|f|pσdx)1pσ≤C(∫Θ|Q∇f(x)|pdx)1p also holds. Central to our proof is showing the existence and boundedness of solutions of the Dirichlet problem{Xp,τu=φ in Θu=0 in ∂Θ, where Xp,τ is a degenerate p-Laplacian operator with a zero order term:Xp,τu=div(|Q∇u|p−2Q∇u)−τ|u|p−2u.

Kernel Treelets

A new method for hierarchical clustering of data points is presented. It combines treelets, a particular multiresolution decomposition of data, with a mapping on a reproducing kernel Hilbert space. The proposed approach, called kernel treelets (KT), uses this mapping to go from a hierarchical clustering over attributes (the natural output of treelets) to a hierarchical clustering over data. KT effectively substitutes the correlation coefficient matrix used in treelets with a symmetric and positive semi-definite matrix efficiently constructed from a symmetric and positive semi-definite kernel function. Unlike most clustering methods, which require data sets to be numeric, KT can be applied to more general data and yields a multiresolution sequence of orthonormal bases on the data directly in feature space. The effectiveness and potential of KT in clustering analysis are illustrated with some examples.