Showing papers on "Bounded function published in 2015"

Leader-Based Optimal Coordination Control for the Consensus Problem of Multiagent Differential Games via Fuzzy Adaptive Dynamic Programming

Abstract: In this paper, a new online scheme is presented to design the optimal coordination control for the consensus problem of multiagent differential games by fuzzy adaptive dynamic programming, which brings together game theory, generalized fuzzy hyperbolic model (GFHM), and adaptive dynamic programming. In general, the optimal coordination control for multiagent differential games is the solution of the coupled Hamilton-Jacobi (HJ) equations. Here, for the first time, GFHMs are used to approximate the solutions (value functions) of the coupled HJ equations, based on policy iteration algorithm. Namely, for each agent, GFHM is used to capture the mapping between the local consensus error and local value function. Since our scheme uses the single-network architecture for each agent (which eliminates the action network model compared with dual-network architecture), it is a more reasonable architecture for multiagent systems. Furthermore, the approximation solution is utilized to obtain the optimal coordination control. Finally, we give the stability analysis for our scheme, and prove the weight estimation error and the local consensus error are uniformly ultimately bounded. Further, the control node trajectory is proven to be cooperative uniformly ultimately bounded.

Global Convergence of Splitting Methods for Nonconvex Composite Optimization

Abstract: We consider the problem of minimizing the sum of a smooth function $h$ with a bounded Hessian and a nonsmooth function. We assume that the latter function is a composition of a proper closed function $P$ and a surjective linear map $\mathcal{M}$, with the proximal mappings of $\tau P$, $\tau > 0$, simple to compute. This problem is nonconvex in general and encompasses many important applications in engineering and machine learning. In this paper, we examined two types of splitting methods for solving this nonconvex optimization problem: the alternating direction method of multipliers and the proximal gradient algorithm. For the direct adaptation of the alternating direction method of multipliers, we show that if the penalty parameter is chosen sufficiently large and the sequence generated has a cluster point, then it gives a stationary point of the nonconvex problem. We also establish convergence of the whole sequence under an additional assumption that the functions $h$ and $P$ are semialgebraic. Further...

Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure

Abstract: In prior work (4) of the first two authors with Savare, a new Riemannian notion of lower bound for Ricci curvature in the class of metric measure spaces (X,d,m) was introduced, and the corresponding class of spaces denoted by RCD(K,∞). This notion relates the CD(K,N) theory of Sturm and Lott-Villani, in the case N = ∞, to the Bakry-Emery approach. In (4) the RCD(K,∞) property is defined in three equivalent ways and several properties of RCD(K,∞) spaces, including the regularization properties of the heat flow, the connections with the theory of Dirichlet forms and the stability under tensor products, are provided. In (4) only finite reference measures m have been considered. The goal of this paper is twofold: on one side we extend these results to general σ-finite spaces, on the other we remove a technical assumption appeared in (4) concerning a strengthening of the CD(K,∞) condition. This more general class of spaces includes Euclidean spaces endowed with Lebesgue measure, complete noncompact Riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with lower Ricci curvature bounds.

A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis

Abstract: The purpose of this work is to study solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions These operators can be realized as the Dirichlet-to-Neumann map for a degenerate/singular elliptic problem posed on a semi-infinite cylinder, which we analyze in the framework of weighted Sobolev spaces Motivated by the rapid decay of the solution to this problem, we propose a truncation that is suitable for numerical approximation We discretize this truncation using first degree tensor product finite elements We derive a priori error estimates in weighted Sobolev spaces The estimates exhibit optimal regularity but suboptimal order for quasi-uniform meshes For anisotropic meshes instead, they are quasi-optimal in both order and regularity We present numerical experiments to illustrate the method's performance

Adaptive Fuzzy Backstepping Control for A Class of Nonlinear Systems With Sampled and Delayed Measurements

Abstract: This paper investigates the adaptive fuzzy backstepping control and ${H_\infty}$ performance analysis for a class of nonlinear systems with sampled and delayed measurements. In the control scheme, a fuzzy-estimator (FE) model is used to estimate the states of the controlled plant, while the fuzzy logic systems are used to approximate the unknown nonlinear functions in the nonlinear system. The controller is obtained based on the FE model by combining the backstepping technique with the classic adaptive fuzzy control method. In the stability analysis, all the signals in the closed-loop system are guaranteed to be semiglobally uniformly ultimately bounded (SUUB) and the outputs of the system are proven to converge to a small neighborhood of origin. Furthermore, the ${H_\infty}$ performance is investigated and the outputs of the closed-loop system are bounded in the ${H_\infty}$ sense. Two examples are given to illustrate the effectiveness of the proposed control scheme.

Global Ill-Posedness of the Isentropic System of Gas Dynamics

Abstract: We consider the isentropic compressible Euler system in 2 space dimensions with pressure law p () = (2) and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1-dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions. (c) 2015 Wiley Periodicals, Inc.

Nonlinear elliptic equations with variable exponent: Old and new

Abstract: In this survey paper, by using variational methods, we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type { − div A ( x , ∇ u ) = λ | u | q ( x ) − 2 u in Ω u = 0 on ∂ Ω , where Ω is a bounded or an exterior domain of R N and q is a continuous positive function. The results presented in this paper extend several contributions concerning the Lane–Emden equation and we focus on new phenomena which are due to the presence of variable exponents.

Finite-Time Cluster Synchronization of T–S Fuzzy Complex Networks With Discontinuous Subsystems and Random Coupling Delays

Abstract: This paper is concerned with the cluster synchronization in finite time for a class of complex networks with nonlinear coupling strengths and probabilistic coupling delays. The complex networks consist of several clusters of nonidentical discontinuous systems suffered from uncertain bounded external disturbance. Based on the Takagi–Sugeno (T–S) fuzzy interpolation approach, we first obtain a set of T–S fuzzy complex networks with constant coupling strengths. By developing some novel Lyapunov functionals and using the concept of Filippov solution, some new analytical techniques are established to derive sufficient conditions ensuring the cluster synchronization in a setting time. In particular, this paper extends the pinning control strategies for networks with continuous-time dynamics to discontinuous networks. Numerical simulations demonstrate that the theoretical results are effective and the T–S fuzzy approach is important for relaxed results.

Learning without Concentration

Abstract: We obtain sharp bounds on the estimation error of the Empirical Risk Minimization procedure, performed in a convex class and with respect to the squared loss, without assuming that class members and the target are bounded functions or have rapidly decaying tails.Rather than resorting to a concentration-based argument, the method used here relies on a “small-ball” assumption and thus holds for classes consisting of heavy-tailed functions and for heavy-tailed targets.The resulting estimates scale correctly with the “noise level” of the problem, and when applied to the classical, bounded scenario, always improve the known bounds.

Convergence Analysis for Anderson Acceleration

Abstract: Anderson($m$) is a method for acceleration of fixed point iteration which stores m+1 prior evaluations of the fixed point map and computes the new iteration as a linear combination of those evaluations. Anderson(0) is fixed point iteration. In this paper we show that Anderson($m$) is locally r-linearly convergent if the fixed point map is a contraction and the coefficients in the linear combination remain bounded. Without assumptions on the coefficients, we prove q-linear convergence of Anderson(1) and, in the case of linear problems, Anderson($m$). We observe that the optimization problem for the coefficients can be formulated and solved in nonstandard ways and report on numerical experiments which illustrate the ideas.

Cooperative Control of Multi-Agent Systems With Unknown State-Dependent Controlling Effects

Abstract: This paper investigates the cooperative control problem of uncertain high-order nonlinear multi-agent systems on directed graph with a fixed topology. Each follower is assumed to have an unknown controlling effect which depends on its own state. By the Nussbaum-type gain technique and the function approximation capability of neural networks, a distributed adaptive neural networks-based controller is designed for each follower in the graph such that all followers can asymptotically synchronize the leader with tracking errors being semi-globally uniform ultimate bounded. Analysis of stability and parameter convergence of the proposed algorithm are conducted based on algebraic graph theory and Lyapunov theory. Finally, a example is provided to validate the theoretical results.

Linearized alternating direction method with parallel splitting and adaptive penalty for separable convex programs in machine learning

Abstract: Many problems in machine learning and other fields can be (re)formulated as linearly constrained separable convex programs. In most of the cases, there are multiple blocks of variables. However, the traditional alternating direction method (ADM) and its linearized version (LADM, obtained by linearizing the quadratic penalty term) are for the two-block case and cannot be naively generalized to solve the multi-block case. So there is great demand on extending the ADM based methods for the multi-block case. In this paper, we propose LADM with parallel splitting and adaptive penalty (LADMPSAP) to solve multi-block separable convex programs efficiently. When all the component objective functions have bounded subgradients, we obtain convergence results that are stronger than those of ADM and LADM, e.g., allowing the penalty parameter to be unbounded and proving the sufficient and necessary conditions for global convergence. We further propose a simple optimality measure and reveal the convergence rate of LADMPSAP in an ergodic sense. For programs with extra convex set constraints, with refined parameter estimation we devise a practical version of LADMPSAP for faster convergence. Finally, we generalize LADMPSAP to handle programs with more difficult objective functions by linearizing part of the objective function as well. LADMPSAP is particularly suitable for sparse representation and low-rank recovery problems because its subproblems have closed form solutions and the sparsity and low-rankness of the iterates can be preserved during the iteration. It is also highly parallelizable and hence fits for parallel or distributed computing. Numerical experiments testify to the advantages of LADMPSAP in speed and numerical accuracy.

Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method

Abstract: We give a new approach to the dictionary learning (also known as "sparse coding") problem of recovering an unknown n x m matrix A (for m ≥ n) from examples of the form [y = Ax + e,] where x is a random vector in Rm with at most τ m nonzero coordinates, and e is a random noise vector in Rn with bounded magnitude For the case m=O(n), our algorithm recovers every column of A within arbitrarily good constant accuracy in time mO(log m/log(τ-1)), in particular achieving polynomial time if τ = m-δ for any δ>0, and time mO(log m) if τ is (a sufficiently small) constant Prior algorithms with comparable assumptions on the distribution required the vector $x$ to be much sparser---at most √n nonzero coordinates---and there were intrinsic barriers preventing these algorithms from applying for denser xWe achieve this by designing an algorithm for noisy tensor decomposition that can recover, under quite general conditions, an approximate rank-one decomposition of a tensor T, given access to a tensor T' that is τ-close to T in the spectral norm (when considered as a matrix) To our knowledge, this is the first algorithm for tensor decomposition that works in the constant spectral-norm noise regime, where there is no guarantee that the local optima of T and T' have similar structuresOur algorithm is based on a novel approach to using and analyzing the Sum of Squares semidefinite programming hierarchy (Parrilo 2000, Lasserre 2001), and it can be viewed as an indication of the utility of this very general and powerful tool for unsupervised learning problems

Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity

Abstract: This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator LK and involving a critical nonlinearity. In particular, we consider the problem −M(||u||2)LKu=λf(x,u)+|u|2s∗−2uin Ω,u=0in Rn∖Ω, where Ω⊂Rn is a bounded domain, 2s∗ is the critical exponent of the fractional Sobolev space Hs(Rn), the function f is a subcritical term and λ is a positive parameter. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function M could be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature.

Randomized Sketches of Convex Programs With Sharp Guarantees

Abstract: Random projection (RP) is a classical technique for reducing storage and computational costs. We analyze RP-based approximations of convex programs, in which the original optimization problem is approximated by solving a lower dimensional problem. Such dimensionality reduction is essential in computation-limited settings, since the complexity of general convex programming can be quite high (e.g., cubic for quadratic programs, and substantially higher for semidefinite programs). In addition to computational savings, RP is also useful for reducing memory usage, and has useful properties for privacy-preserving optimization. We prove that the approximation ratio of this procedure can be bounded in terms of the geometry of the constraint set. For a broad class of RPs, including those based on various sub-Gaussian distributions as well as randomized Hadamard and Fourier transforms, the data matrix defining the cost function can be projected to a dimension proportional to the squared Gaussian width of the tangent cone of the constraint set at the original solution. This effective dimension of the convex program is often substantially smaller than the original dimension. We illustrate consequences of our theory for various cases, including unconstrained and $\ell _{1}$ -constrained least squares, support vector machines, low-rank matrix estimation, and discuss implications for privacy-preserving optimization, as well as connections with denoising and compressed sensing.

The boundedness-by-entropy method for cross-diffusion systems

Abstract: The global-in-time existence of bounded weak solutions to a large class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure is proved. The main feature of these systems is that the diffusion matrix may be generally neither symmetric nor positive semi-definite. The key idea is to employ a transformation of variables, determined by the entropy density, which is defined by the gradient-flow formulation. The transformation yields at the same time a positive semi-definite diffusion matrix, suitable gradient estimates as well as lower and/or upper bounds of the solutions. These bounds are a consequence of the transformation of variables and are obtained without the use of a maximum principle. Several classes of cross-diffusion systems are identified which can be solved by this technique. The systems are formally derived from continuous-time random walks on a lattice modeling, for instance, the motion of ions, cells, or fluid particles. The key conditions for this approach are identified and previous results in the literature are unified and generalized. New existence results are obtained for the population model with or without volume filling.

Adaptive NN Controller Design for a Class of Nonlinear MIMO Discrete-Time Systems

Abstract: An adaptive neural network tracking control is studied for a class of multiple-input multiple-output (MIMO) nonlinear systems. The studied systems are in discrete-time form and the discretized dead-zone inputs are considered. In addition, the studied MIMO systems are composed of $N$ subsystems, and each subsystem contains unknown functions and external disturbance. Due to the complicated framework of the discrete-time systems, the existence of the dead zone and the noncausal problem in discrete-time, it brings about difficulties for controlling such a class of systems. To overcome the noncausal problem, by defining the coordinate transformations, the studied systems are transformed into a special form, which is suitable for the backstepping design. The radial basis functions NNs are utilized to approximate the unknown functions of the systems. The adaptation laws and the controllers are designed based on the transformed systems. By using the Lyapunov method, it is proved that the closed-loop system is stable in the sense that the semiglobally uniformly ultimately bounded of all the signals and the tracking errors converge to a bounded compact set. The simulation examples and the comparisons with previous approaches are provided to illustrate the effectiveness of the proposed control algorithm.

Gauss-Bonnet inflation

Abstract: We consider an Einstein-scalar-Gauss-Bonnet gravitational theory, and argue that at early times the Ricci scalar can be safely ignored. We then demonstrate that the pure scalar-Gauss-Bonnet theory, with a quadratic coupling function, naturally supports inflationary---de Sitter---solutions. During inflation, the scalar field decays exponentially and its effective potential remains always bounded. The theory also contains solutions where these de Sitter phases possess a natural exit mechanism and are replaced by linearly expanding---Milne---phases.

Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains

Abstract: We consider nonlinear diffusive evolution equations posed on bounded space domains, governed by fractional Laplace-type operators, and involving porous medium type nonlinearities. We establish existence and uniqueness results in a suitable class of solutions using the theory of maximal monotone operators on dual spaces. Then we describe the long-time asymptotics in terms of separate-variables solutions of the friendly giant type. As a by-product, we obtain an existence and uniqueness result for semilinear elliptic non local equations with sub-linear nonlinearities. The Appendix contains a review of the theory of fractional Sobolev spaces and of the interpolation theory that are used in the rest of the paper.

New class of chaotic systems with circular equilibrium

Abstract: This paper brings a new mathematical model of the third-order autonomous deterministic dynamical system with associated chaotic motion. Its unique property lies in the existence of circular equilibrium which was not, by referring to the best knowledge of the authors, so far reported. Both mathematical analysis and circuitry implementation of the corresponding differential equations are presented. It is shown that discovered system provides a structurally stable strange attractor which fulfills fractal dimensionality and geometrical density and is bounded into a finite state space volume.

Containment control of linear multi-agent systems with multiple leaders of bounded inputs using distributed continuous controllers

Abstract: Summary This paper considers the containment control problem for multi-agent systems with general linear dynamics and multiple leaders whose control inputs are possibly nonzero and time varying. Based on the relative states of neighboring agents, a distributed static continuous controller is designed, under which the containment error is uniformly ultimately bounded and the upper bound of the containment error can be made arbitrarily small, if the subgraph associated with the followers is undirected and, for each follower, there exists at least one leader that has a directed path to that follower. It is noted that the design of the static controller requires the knowledge of the eigenvalues of the Laplacian matrix and the upper bounds of the leaders’ control inputs. In order to remove these requirements, a distributed adaptive continuous controller is further proposed, which can be designed and implemented by each follower in a fully distributed fashion. Extensions to the case where only local output information is available and to the case of multi-agent systems with matching uncertainties are also discussed. Copyright © 2014 John Wiley & Sons, Ltd.

Algorithmic Stability for Adaptive Data Analysis

Abstract: Adaptivity is an important feature of data analysis---the choice of questions to ask about a dataset often depends on previous interactions with the same dataset. However, statistical validity is typically studied in a nonadaptive model, where all questions are specified before the dataset is drawn. Recent work by Dwork et al. (STOC, 2015) and Hardt and Ullman (FOCS, 2014) initiated the formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis. Specifically, suppose there is an unknown distribution $\mathbf{P}$ and a set of $n$ independent samples $\mathbf{x}$ is drawn from $\mathbf{P}$. We seek an algorithm that, given $\mathbf{x}$ as input, accurately answers a sequence of adaptively chosen queries about the unknown distribution $\mathbf{P}$. How many samples $n$ must we draw from the distribution, as a function of the type of queries, the number of queries, and the desired level of accuracy? In this work we make two new contributions: (i) We give upper bounds on the number of samples $n$ that are needed to answer statistical queries. The bounds improve and simplify the work of Dwork et al. (STOC, 2015), and have been applied in subsequent work by those authors (Science, 2015, NIPS, 2015). (ii) We prove the first upper bounds on the number of samples required to answer more general families of queries. These include arbitrary low-sensitivity queries and an important class of optimization queries. As in Dwork et al., our algorithms are based on a connection with algorithmic stability in the form of differential privacy. We extend their work by giving a quantitatively optimal, more general, and simpler proof of their main theorem that stability implies low generalization error. We also study weaker stability guarantees such as bounded KL divergence and total variation distance.

Regularity of Einstein manifolds and the codimension 4 conjecture

Abstract: In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds (M n ;g) with bounded Ricci curvature, as well as their Gromov-Hausdor limit spaces ( M n ;dj) dGH ! (X;d), where dj denotes the Riemannian distance. Our main result is a solution to the codimension 4 conjecture, namely thatX is smooth away from a closed subset of codimension 4. We combine this result with the ideas of quantitative stratication to prove a priori L q estimates on the full curvaturejRmj for all q v > 0, and diam(M) D contains at most a nite number of dieomorphism classes. A local version is used to show that noncollapsed 4-manifolds with bounded Ricci curvature have a priori L 2 Riemannian curvature estimates.

Clustered Integer 3SUM via Additive Combinatorics

Abstract: We present a collection of new results on problems related to 3SUM, including: The first truly subquadratic algorithm for computing the (min,+) convolution for monotone increasing sequences with integer values bounded by O(n), solving 3SUM for monotone sets in 2D with integer coordinates bounded by O(n), and preprocessing a binary string for histogram indexing (also called jumbled indexing).The running time is O(n(9+√177)/12, polylog,n)=O(n1.859) with randomization, or O(n1.864) deterministically. This greatly improves the previous n2/2Ω(√log n) time bound obtained from Williams' recent result on all-pairs shortest paths [STOC'14], and answers an open question raised by several researchers studying the histogram indexing problem. The first algorithm for histogram indexing for any constant alphabet size that achieves truly subquadratic preprocessing time and truly sublinear query time. A truly subquadratic algorithm for integer 3SUM in the case when the given set can be partitioned into n1-δ clusters each covered by an interval of length n, for any constant δ>0. An algorithm to preprocess any set of n integers so that subsequently 3SUM on any given subset can be solved in O(n13/7, polylog,n) time.All these results are obtained by a surprising new technique, based on the Balog--Szemeredi--Gowers Theorem from additive combinatorics.

Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity

Abstract: We consider a chemotaxis-fluid system involving nonlinear cell diffusion of porous medium type, signal consumption by cells, and rather general, possibly matrix-valued, chemotactic sensitivities. It is shown that if the corresponding diffusion exponent $m$ satisfies $m>7/6$, then for all reasonably regaular initial data an associated initial-boundary value problem in smoothly bounded three-dimensional domains possesses a globally defined weak solution which is bounded. Under a mild additional assumption on the signal consumption rate, it is moreover shown that any nontrivial of these solutions stabilizes toward a spatially homogeneous equilibrium in the large time limit.