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Showing papers on "Bounded function published in 2008"


Journal ArticleDOI
Wei Ren1
TL;DR: This note shows that consensus is reached asymptotically for the first two cases if the undirected interaction graph is connected and for the third case if the directed interaction graph has a directed spanning tree and the gain for velocity matching with the group reference velocity is above a certain bound.
Abstract: This note considers consensus algorithms for double-integrator dynamics. We propose and analyze consensus algorithms for double-integrator dynamics in four cases: 1) with a bounded control input, 2) without relative velocity measurements, 3) with a group reference velocity available to each team member, and 4) with a bounded control input when a group reference state is available to only a subset of the team. We show that consensus is reached asymptotically for the first two cases if the undirected interaction graph is connected. We further show that consensus is reached asymptotically for the third case if the directed interaction graph has a directed spanning tree and the gain for velocity matching with the group reference velocity is above a certain bound. We also show that consensus is reached asymptotically for the fourth case if and only if the group reference state flows directly or indirectly to all of the vehicles in the team.

1,338 citations


Journal ArticleDOI
TL;DR: It is proved that any bounded sequence generated by the proximal algorithm converges to some generalized critical point and the decay estimates that are derived are of the type O(k−s), where s ∈ (0, + ∞) depends on the flatness of the function.
Abstract: We study the convergence of the proximal algorithm applied to nonsmooth functions that satisfy the Łjasiewicz inequality around their generalized critical points. Typical examples of functions complying with these conditions are continuous semialgebraic or subanalytic functions. Following Łjasiewicz’s original idea, we prove that any bounded sequence generated by the proximal algorithm converges to some generalized critical point. We also obtain convergence rate results which are related to the flatness of the function by means of Łjasiewicz exponents. Apart from the sharp and elliptic cases which yield finite or geometric convergence, the decay estimates that are derived are of the type O(k −s ), where s ∈ (0, + ∞) depends on the flatness of the function.

590 citations


Posted Content
TL;DR: A convergent proximal reweighted l1 algorithm for compressive sensing and an application to rank reduction problems is provided, which depends on the geometrical properties of the function L around its critical points.
Abstract: We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: $L(x,y)=f(x)+Q(x,y)+g(y)$, where $f:\R^n\rightarrow\R\cup{+\infty}$ and $g:\R^m\rightarrow\R\cup{+\infty}$ are proper lower semicontinuous functions, and $Q:\R^n\times\R^m\rightarrow \R$ is a smooth $C^1$ function which couples the variables $x$ and $y$. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize $L$. We work in a nonconvex setting, just assuming that the function $L$ satisfies the Kurdyka-\L ojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to "metrically regular" problems. Our main result can be stated as follows: If L has the Kurdyka-\L ojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of $L$. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function $L$ around its critical points. When specialized to $Q(x,y)=|x-y|^2$ and to $f$, $g$ indicator functions, the algorithm is an alternating projection mehod (a variant of Von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with "regular" intersection. In order to illustrate our results with concrete problems, we provide a convergent proximal reweighted $\ell^1$ algorithm for compressive sensing and an application to rank reduction problems.

569 citations


Journal ArticleDOI
TL;DR: In this article, the authors describe a connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K3 surface. But their analysis is restricted to the case where the stable sheaves are coherent.
Abstract: This article contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K3 surface

525 citations


Proceedings ArticleDOI
25 Oct 2008
TL;DR: In this article, a stream-cipher S whose implementation is secure even if a bounded amount of arbitrary (adversarially chosen) information on the internal state ofS is leaked during computation is presented.
Abstract: We construct a stream-cipher S whose implementation is secure even if a bounded amount of arbitrary (adversarially chosen) information on the internal state ofS is leaked during computation. This captures all possible side-channel attacks on S where the amount of information leaked in a given period is bounded, but overall can be arbitrary large. The only other assumption we make on the implementation of S is that only data that is accessed during computation leaks information. The stream-cipher S generates its output in chunks K1, K2, . . . and arbitrary but bounded information leakage is modeled by allowing the adversary to adaptively chose a function fl : {0,1}* rarr {0, 1}lambda before Kl is computed, she then gets fl(taul) where taul is the internal state ofS that is accessed during the computation of Kg. One notion of security we prove for S is that Kg is indistinguishable from random when given K1,..., K1-1,f1(tau1 ),..., fl-1(taul-1) and also the complete internal state of S after Kg has been computed (i.e. S is forward-secure). The construction is based on alternating extraction (used in the intrusion-resilient secret-sharing scheme from FOCS'07). We move this concept to the computational setting by proving a lemma that states that the output of any PRG has high HILLpseudoentropy (i.e. is indistinguishable from some distribution with high min-entropy) even if arbitrary information about the seed is leaked. The amount of leakage lambda that we can tolerate in each step depends on the strength of the underlying PRG, it is at least logarithmic, but can be as large as a constant fraction of the internal state of S if the PRG is exponentially hard.

519 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider a class of zero-sum two-player stochastic games called tug-of-war and use them to prove that every bounded real-valued Lipschitz function F on a subset Y of a length space X admits a unique AM extension to X.
Abstract: We consider a class of zero-sum two-player stochastic games called tug-of-war and use them to prove that every bounded real-valued Lipschitz function F on a subset Y of a length space X admits a unique absolutely minimal (AM) extension to X, i.e., a unique Lipschitz extension u : X → ℝ for which Lip U u = Lip ∂u u for all open U ⊂ X \ Y.

438 citations


Journal ArticleDOI
TL;DR: A new realization of the fourth-order derivative Pais-Uhlenbeck oscillator is constructed that possesses no states of negative norm and has a real energy spectrum that is bounded below.
Abstract: A new realization of the fourth-order derivative Pais-Uhlenbeck oscillator is constructed. This realization possesses no states of negative norm and has a real energy spectrum that is bounded below. The key to this construction is the recognition that in this realization the Hamiltonian is not Dirac Hermitian. However, the Hamiltonian is symmetric under combined space reflection P and time reversal T. The Hilbert space that is appropriate for this PT-symmetric Hamiltonian is identified and it is found to have a positive-definite inner product. Furthermore, the time-evolution operator is unitary.

373 citations


Book ChapterDOI
01 Jan 2008
TL;DR: In this paper, the authors considered the problem of dissipative dynamical systems in unbounded domains and showed that the dynamics generated by dissipative PDEs in such domains are purely infinite dimensional and do not possess any finite dimensional reduction principle.
Abstract: Publisher Summary The study of the asymptotic behavior of dynamical systems arising from mechanics and physics is a capital issue because it is essential for practical applications to be able to understand and even predict the long time behavior of the solutions of such systems. A dynamical system is a (deterministic) system that evolves with respect to the time. Such a time evolution can be continuous or discrete (i.e., the state of the system is measured only at given times, for example, every hour or every day). The chapter essentially considers continuous dynamical systems. While the theory of attractors for dissipative dynamical systems in bounded domains is rather well understood, the situation is different for systems in unbounded domains and such a theory has only recently been addressed (and is still progressing), starting from the pioneering works of Abergel and Babin and Vishik. The main difficulty in this theory is the fact that, in contrast to the case of bounded domains discussed above, the dynamics generated by dissipative PDEs in unbounded domains is (as a rule) purely infinite dimensional and does not possess any finite dimensional reduction principle. In addition, the additional spatial “unbounded” directions lead to the so-called spatial chaos and the interactions between spatial and temporal chaotic modes generate a space–time chaos, which also has no analogue in finite dimensions.

321 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that a slice of the union of tight geodesics between any pair of points has cardinality bounded purely in terms of the topological type of the curve complex.
Abstract: The curve graph, \(\mathcal{G}\), associated to a compact surface Σ is the 1-skeleton of the curve complex defined by Harvey. Masur and Minsky showed that this graph is hyperbolic and defined the notion of a tight geodesic therein. We prove some finiteness results for such geodesics. For example, we show that a slice of the union of tight geodesics between any pair of points has cardinality bounded purely in terms of the topological type of Σ. We deduce some consequences for the action of the mapping class group on \(\mathcal{G}\). In particular, we show that it satisfies an acylindricity condition, and that the stable lengths of pseudoanosov elements are rational with bounded denominator.

318 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that for any η > 0, the (2 + η)th-moment of a complex random variable is bounded, under the slightly stronger assumption that the second-order moments of the variable are bounded.
Abstract: Let x be a complex random variable with mean zero and bounded variance σ2. Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ1, …, λn be the eigenvalues of . Define the empirical spectral distributionμn of Nn by the formula The following well-known conjecture has been open since the 1950's: Circular Law Conjecture: μn converges to the uniform distribution μ∞ over the unit disk as n tends to infinity. We prove this conjecture, with strong convergence, under the slightly stronger assumption that the (2 + η)th-moment of x is bounded, for any η > 0. Our method builds and improves upon earlier work of Girko, Bai, Gotze–Tikhomirov, and Pan–Zhou, and also applies for sparse random matrices. The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.

304 citations


Journal ArticleDOI
TL;DR: It is shown that, under mild assumptions and for large N, the occupancy measure converges, in mean square (and thus in probability) over any finite horizon, to a deterministic dynamical system.

Journal ArticleDOI
TL;DR: This work gives a new sufficient condition on the boundary conditions for the exponential stability of one-dimensional nonlinear hyperbolic systems on a bounded interval using an explicit strict Lyapunov function.
Abstract: We give a new sufficient condition on the boundary conditions for the exponential stability of one-dimensional nonlinear hyperbolic systems on a bounded interval. Our proof relies on the construction of an explicit strict Lyapunov function. We compare our sufficient condition with other known sufficient conditions for nonlinear and linear one-dimensional hyperbolic systems.

Journal ArticleDOI
01 Feb 2008
TL;DR: An adaptive backstepping neural-network control approach is extended to a class of large-scale nonlinear output-feedback systems with completely unknown and mismatched interconnections to remove the common assumptions on interconnection such as matching condition, bounded by upper bounding functions.
Abstract: An adaptive backstepping neural-network control approach is extended to a class of large-scale nonlinear output-feedback systems with completely unknown and mismatched interconnections. The novel contribution is to remove the common assumptions on interconnections such as matching condition, bounded by upper bounding functions. Differentiation of the interconnected signals in backstepping design is avoided by replacing the interconnected signals in neural inputs with the reference signals. Furthermore, two kinds of unknown modeling errors are handled by the adaptive technique. All the closed-loop signals are guaranteed to be semiglobally uniformly ultimately bounded, and the tracking errors are proved to converge to a small residual set around the origin. The simulation results illustrate the effectiveness of the control approach proposed in this correspondence.

Posted Content
TL;DR: Hodge decomposition sheds light on whether a given dataset may be globally ranked in a meaningful way or if the data is inherently inconsistent and thus could not have any reasonable global ranking; in the latter case it provides information on the nature of the inconsistencies.
Abstract: We propose a number of techniques for obtaining a global ranking from data that may be incomplete and imbalanced -- characteristics almost universal to modern datasets coming from e-commerce and internet applications. We are primarily interested in score or rating-based cardinal data. From raw ranking data, we construct pairwise rankings, represented as edge flows on an appropriate graph. Our statistical ranking method uses the graph Helmholtzian, the graph theoretic analogue of the Helmholtz operator or vector Laplacian, in much the same way the graph Laplacian is an analogue of the Laplace operator or scalar Laplacian. We study the graph Helmholtzian using combinatorial Hodge theory: we show that every edge flow representing pairwise ranking can be resolved into two orthogonal components, a gradient flow that represents the L2-optimal global ranking and a divergence-free flow (cyclic) that measures the validity of the global ranking obtained -- if this is large, then the data does not have a meaningful global ranking. This divergence-free flow can be further decomposed orthogonally into a curl flow (locally cyclic) and a harmonic flow (locally acyclic but globally cyclic); these provides information on whether inconsistency arises locally or globally. An obvious advantage over the NP-hard Kemeny optimization is that discrete Hodge decomposition may be computed via a linear least squares regression. We also investigated the L1-projection of edge flows, showing that this is dual to correlation maximization over bounded divergence-free flows, and the L1-approximate sparse cyclic ranking, showing that this is dual to correlation maximization over bounded curl-free flows. We discuss relations with Kemeny optimization, Borda count, and Kendall-Smith consistency index from social choice theory and statistics.

Journal ArticleDOI
TL;DR: A generalized least-squares approach that consists in minimizing a quadratic estimation cost function defined on a recent batch of inputs and outputs according to a sliding-window strategy is used, and the existence of bounding sequences on the estimation error is proved.

Journal ArticleDOI
TL;DR: For general bounded domains Ω and resolutive functions F, this paper showed that for sufficiently regular Ω, the functions ue converge uniformly to the unique p-harmonic extension of F and showed that the game ends when the game position reaches some y∈∂Ω, and player I's payoff is F(y).
Abstract: Fix a bounded domain Ω⊂Rd, a continuous function F:∂Ω→R, and constants e>0 and 1

Journal ArticleDOI
TL;DR: It is shown that, as long as the hidden layer activation function is complex continuous discriminatory or complex bounded nonlinear piecewise continuous, I-ELM can still approximate any target functions in the complex domain.

Journal Article
TL;DR: In this paper, the authors considered the wave equation in a bounded region with a smooth boundary with distributed delay on the boundary or into the domain, and proved the expo- nential stability of the solution.
Abstract: We consider the wave equation in a bounded region with a smooth boundary with distributed delay on the boundary or into the domain. In both cases, under suitable assumptions, we prove the expo- nential stability of the solution. These results are obtained by introduc- ing suitable energies and by proving some observability inequalities. For an internal distributed delay, we further show some instability results.

Journal ArticleDOI
TL;DR: In this article, the authors consider an elliptic-parabolic system of the Keller-Segel type with nonlinear diffusion and find a critical exponent of the nonlinearity in the diffusion, measuring the strength of diffusion at points of high population densities.
Abstract: We consider an elliptic–parabolic system of the Keller–Segel type which involves nonlinear diffusion. We find a critical exponent of the nonlinearity in the diffusion, measuring the strength of diffusion at points of high (population) densities, which distinguishes between finite-time blow-up and global-in-time existence of uniformly bounded solutions. This critical exponent depends on the space dimension n ≥ 1, where apart from the physically relevant cases n = 2 and n = 3 also the result obtained in the one-dimensional setting might be of mathematical interest: here, namely, finite-time explosion of solutions occurs although the Lyapunov functional associated with the system is bounded from below. Additionally this one-dimensional case is an example to show that L∞ estimates of solutions to non-uniformly parabolic drift–diffusion equations cannot be expected even when boundedness of the gradient of the drift term is presupposed.

Journal ArticleDOI
TL;DR: In this paper, it was shown that a multiplicative stochastic perturbation of Brownian type is enough to render the linear transport equation well-posed under the influece of noise.
Abstract: We consider the linear transport equation with a globally Holder continuous and bounded vector field. While this deterministic PDE may not be well-posed, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influece of noise. The key tool is a differentiable stochastic flow constructed and analysed by means of a special transformation of the drift of Ito-Tanaka type.

Journal ArticleDOI
TL;DR: In this article, the existence, uniqueness, and space-time Holder regular-ity for solutions of the parabolic stochastic evolution equation dU(t) = (AU(t)) + F(t,U (t)))dt + B(t and U(t)), t 2 (0,T0), U(0) = u0, where A generates an analytic C0-semigroup on a UMD Banach space E and WH is a cylindrical Brownian motion with values in a Hilbert space H.

Journal ArticleDOI
TL;DR: LMI-based synthesis tools for regional stability and performance of linear anti-windup compensators for linear control systems are presented and it is shown that for systems whose plants have poles in the closed left-half plane, plant-order dynamic anti- windup can achieve semiglobal exponential stability and finite L"2 gain for exogenous inputs with L" 2 norm bounded by any finite value.

Journal ArticleDOI
TL;DR: It is proved that nonconservative schemes generate, at the level of the limiting hyperbolic system, an convergence error source-term which, provided the total variation of the approximations remains uniformly bounded, is a locally bounded measure.

Journal ArticleDOI
TL;DR: It is demonstrated that a weak value can be nonclassical if and only if a Leggett-Garg inequality can also be violated, and generalized weak values are described in which post-selection occurs on a range of weak measurement results.
Abstract: An implementation of weak values is investigated in solid-state qubits. We demonstrate that a weak value can be nonclassical if and only if a Leggett-Garg inequality can also be violated. Generalized weak values are described in which post-selection occurs on a range of weak measurement results. Imposing classical weak values permits the derivation of Leggett-Garg inequalities for bounded operators. Our analysis is presented in terms of kicked quantum nondemolition measurements on a quantum double-dot charge qubit.

Book ChapterDOI
TL;DR: It is proved that the maximum of the sample importance weights in a high-dimensional Gaussian particle filter converges to unity unless the ensemble size grows exponentially in the system dimension, which essentially permits the effective system dimension to be bounded.
Abstract: We prove that the maximum of the sample importance weights in a high-dimensional Gaussian particle filter converges to unity unless the ensemble size grows exponentially in the system dimension. Our work is motivated by and parallels the derivations of Bengtsson, Bickel and Li (2007); however, we weaken their assumptions on the eigenvalues of the covariance matrix of the prior distribution and establish rigorously their strong conjecture on when weight collapse occurs. Specifically, we remove the assumption that the nonzero eigenvalues are bounded away from zero, which, although the dimension of the involved vectors grow to infinity, essentially permits the effective system dimension to be bounded. Moreover, with some restrictions on the rate of growth of the maximum eigenvalue, we relax their assumption that the eigenvalues are bounded from above, allowing the system to be dominated by a single mode.

Book ChapterDOI
Elad Hazan1
07 Apr 2008
TL;DR: An algorithm for approximately maximizing a concave function over the bounded semi-definite cone, which produces sparse solutions for SDP, and a linear time algorithm for Quantum State Tomography is derived.
Abstract: We propose an algorithm for approximately maximizing a concave function over the bounded semi-definite cone, which produces sparse solutions. Sparsity for SDP corresponds to low rank matrices, and is a important property for both computational as well as learning theoretic reasons. As an application, building on Aaronson's recent work, we derive a linear time algorithm for Quantum State Tomography.

Journal ArticleDOI
TL;DR: This work proves the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs in boundeddegree graphs for the Minimum Dominating Set problem.
Abstract: We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs. Using a similar result obtained by Trevisan for Minimum Set Cover we prove the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs. Asymptotically, for degree bound approaching infinity, these bounds almost match the known upper bounds. The results are applied to improve the lower bounds for other related problems such as Maximum Induced Matching and Maximum Leaf Spanning Tree.

Proceedings Article
08 Dec 2008
TL;DR: In this article, the authors consider a generalization of stochastic bandit problems where the set of arms, Χ, is allowed to be a generic topological space and constraint the mean-payoff function with a dissimilarity function over Χ in a way that is more general than Lipschitz.
Abstract: We consider a generalization of stochastic bandit problems where the set of arms, Χ, is allowed to be a generic topological space. We constraint the mean-payoff function with a dissimilarity function over Χ in a way that is more general than Lipschitz. We construct an arm selection policy whose regret improves upon previous result for a large class of problems. In particular, our results imply that if Χ is the unit hypercube in a Euclidean space and the mean-payoff function has a finite number of global maxima around which the behavior of the function is locally Holder with a known exponent, then the expected regret is bounded up to a logarithmic factor by √n, i.e., the rate of the growth of the regret is independent of the dimension of the space. Moreover, we prove the minimax optimality of our algorithm for the class of mean-payoff functions we consider.

Journal ArticleDOI
TL;DR: A new entropy-like inequality introduced by Bresch and Desjardins for the shallow water system of equations gives additional regularity for the density (provided such regularity exists at initial time) and it is proved that the solution is unique in the class of weak solutions satisfying the usual entropy inequality.
Abstract: We consider Navier–Stokes equations for compressible viscous fluids in one dimension. It is a well-known fact that if the initial datum are smooth and the initial density is bounded by below by a positive constant, then a strong solution exists locally in time. In this paper, we show that under the same hypothesis, the density remains bounded by below by a positive constant uniformly in time, and that strong solutions therefore exist globally in time. Moreover, while most existence results are obtained for positive viscosity coefficients, the present result holds even if the viscosity coefficient vanishes with the density. Finally, we prove that the solution is unique in the class of weak solutions satisfying the usual entropy inequality. The key point of the paper is a new entropy-like inequality introduced by Bresch and Desjardins for the shallow water system of equations. This inequality gives additional regularity for the density (provided such regularity exists at initial time).

Journal ArticleDOI
TL;DR: A parameter estimation routine that allows exact reconstruction of the unknown parameters in finite time provided a given excitation condition is satisfied and the algorithm is independent of the control and identifier structure employed.
Abstract: This note presents a parameter estimation routine that allows exact reconstruction of the unknown parameters in finite time provided a given excitation condition is satisfied. The robustness of the routine to an unknown bounded disturbance or modeling error is also shown. The result is independent of the control and identifier structures employed. The true parameter value is obtained without requiring the measurement or computation of the velocity state vector. Moreover, the technique provides a direct solution to the problem of removing auxiliary perturbation signals when parameter convergence is achieved.