Showing papers on "Bounded function published in 2020"

Lectures on von Neumann Algebras

Abstract: Written in lucid language, this valuable text discusses fundamental concepts of von Neumann algebras including bounded linear operators in Hilbert spaces, finite von Neumann algebras, linear forms on algebra of operators, geometry of projections and classification of von Neumann algebras in an easy to understand manner. The revised text covers new material including the first two examples of factors of type II^1, an example of factor of type III and theorems for von Neumann algebras with a cyclic and separating vector. Pedagogical features including solved problems and exercises are interspersed throughout the book.

Consensus Tracking Control of Switched Stochastic Nonlinear Multiagent Systems via Event-Triggered Strategy

Abstract: In this paper, the consensus tracking problem is investigated for a class of continuous switched stochastic nonlinear multiagent systems with an event-triggered control strategy. For continuous stochastic multiagent systems via event-triggered protocols, it is rather difficult to avoid the Zeno behavior by the existing methods. Thus, we propose a new protocol design framework for the underlying systems. It is proven that follower agents can almost surely track the given leader signal with bounded errors and no agent exhibits the Zeno behavior by the given control scheme. Finally, two numerical examples are given to illustrate the effectiveness and advantages of the new design techniques.

Observed-based Adaptive Finite-Time Tracking Control for a Class of Nonstrict-Feedback Nonlinear Systems with Input Saturation

Abstract: This paper concentrates upon the problem of adaptive neural finite-time tracking control for uncertain nonstrict-feedback nonlinear systems with input saturation. The design difficulty of non-smooth input saturation nonlinearity is solved by applying a smooth non-affine function to approximate the saturation signal. Neural networks, as a kind of specialized function estimators, are used to estimate the uncertain function. Meanwhile, a neural network-based observer is constructed to observe the unavailable states, and thus an observer-based adaptive finite-time tracking control strategy is developed by combining dynamic surface control (DSC) technique and backstepping approach. Furthermore, the stability of the considered system is analyzed via semi-global practical finite-time stability theory. Under the proposed control method, all the signals in the closed-loop system are bounded, and the system output can almost surely track the desired trajectory within a specified bounded error in a finite time. In the end, two examples are adopted to illustrate the validity of our results.

Event-Triggered and Self-Triggered $H_\infty$ Control of Uncertain Switched Linear Systems

Abstract: This paper studies the problem of event-triggered and self-triggered $\boldsymbol {H}_{\boldsymbol {\infty }}$ controls for uncertain switched linear systems with exogenous disturbances whose magnitude is bounded by the system state’s norm. With the proposed schemes, the control task is carried out only when the triggering condition is met. That leads to changing and adaptive interexecution intervals and can further reduce the system resource costs to a certain extent. Moreover, to ensure the $\boldsymbol {H}_{\boldsymbol {\infty }}$ control performance of the event-triggered switched system, a proof which makes certain that the occurrence of Zeno problem can be prevented is first provided. Then, by adopting the average dwell time method, a set of sufficient conditions for $\boldsymbol {H}_{\boldsymbol {\infty }}$ performance analysis is developed. Subsequently, the codesign of controller gains and event-triggering schemes is provided. On the basis of the event-triggered control, a solution to the self-triggered control problem is then presented. It can only utilize the current sampled data to predict the next triggered instant. Finally, in order to test the effectiveness of the proposed methods, numerical simulations are performed.

Event-Triggered Adaptive Dynamic Programming for Zero-Sum Game of Partially Unknown Continuous-Time Nonlinear Systems

Abstract: In this paper, the zero-sum game problem is considered for partially unknown continuous-time nonlinear systems, and an event-triggered adaptive dynamic programming (ADP) method is developed to solve the problem. First, an identifier neural network (NN) and a critic NN are applied to approximate the drift system dynamics and the optimal value function, respectively. Subsequently, an event-triggered approach is developed based on ADP, which samples the states and updates the weights of NNs at the same time when the event-triggering condition is violated, such that the computational complexity is reduced. It is proved that the states and the error of NN weights are uniformly ultimately bounded. Finally, the effectiveness of the developed ADP-based event-triggered method is verified through simulation studies.

Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates

Abstract: The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show for generic metrics on a 3-manifold, minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen-Cahn setting, a strong form of the multiplicity one conjecture and the index lower bound conjecture of Marques-Neves in 3-dimensions regarding min-max constructions of minimal surfaces. Allen-Cahn min-max constructions were recently carried out by Guaraco and Gaspar-Guaraco. Our resolution of the multiplicity one and the index lower bound conjectures shows that these constructions can be applied to give a new proof of Yau's conjecture on infinitely many minimal surfaces in a 3-manifold with a generic metric (recently proven by Irie-Marques-Neves) with new geometric conclusions. Namely, we prove that a 3-manifold with a generic metric contains, for every $p$ = 1, 2, 3, ..., a two-sided embedded minimal surface with Morse index $p$ and area $p^{1/3}$, as conjectured by Marques-Neves.

Fluctuation-response inequality out of equilibrium.

Abstract: We present an approach to response around arbitrary out-of-equilibrium states in the form of a fluctuation–response inequality (FRI). We study the response of an observable to a perturbation of the underlying stochastic dynamics. We find that the magnitude of the response is bounded from above by the fluctuations of the observable in the unperturbed system and the Kullback–Leibler divergence between the probability densities describing the perturbed and the unperturbed system. This establishes a connection between linear response and concepts of information theory. We show that in many physical situations, the relative entropy may be expressed in terms of physical observables. As a direct consequence of this FRI, we show that for steady-state particle transport, the differential mobility is bounded by the diffusivity. For a “virtual” perturbation proportional to the local mean velocity, we recover the thermodynamic uncertainty relation (TUR) for steady-state transport processes. Finally, we use the FRI to derive a generalization of the uncertainty relation to arbitrary dynamics, which involves higher-order cumulants of the observable. We provide an explicit example, in which the TUR is violated but its generalization is satisfied with equality.

Delayed Impulsive Control for Consensus of Multiagent Systems With Switching Communication Graphs

Abstract: Delayed impulsive controllers are proposed in this paper to enable the agents in a class of second-order multiagent systems (MASs) to achieve state consensus, based, respectively, on the relative full-state and partial-state sampled-data measurements among neighboring agents. It is a challenging task to analyze the consensus behaviors of the considered MASs as the dynamics of such MASs will be subjected to joint effects from delay-dependent impulses, aperiodic sampling, and switchings among different communication graphs. A novel analytical approach, based upon the discretization method, state augmentation, and linear state transformation, is developed to establish the sufficient consensus criteria on the range of the impulsive intervals and the control parameters. Remarkably, it is found that consensus in the closed-loop MASs can be always ensured by skillfully selecting the control parameters as long as the nonuniform delays and the impulsive intervals are bounded. A numerical example is finally performed to validate the effectiveness of the proposed delayed impulsive controllers.

Adaptive Neural Event-Triggered Control for Discrete-Time Strict-Feedback Nonlinear Systems

Abstract: This paper proposes a novel event-triggered (ET) adaptive neural control scheme for a class of discrete-time nonlinear systems in a strict-feedback form. In the proposed scheme, the ideal control input is derived in a recursive design process, which relies on system states only and is unrelated to virtual control laws. In this case, the high-order neural networks (NNs) are used to approximate the ideal control input (but not the virtual control laws), and then the corresponding adaptive neural controller is developed under the ET mechanism. A modified NN weight updating law, nonperiodically tuned at triggering instants, is designed to guarantee the uniformly ultimate boundedness (UUB) of NN weight estimates for all sampling times. In virtue of the bounded NN weight estimates and a dead-zone operator, the ET condition together with an adaptive ET threshold coefficient is constructed to guarantee the UUB of the closed-loop networked control system through the Lyapunov stability theory, thereby largely easing the network communication load. The proposed ET condition is easy to implement because of the avoidance of: 1) the use of the intermediate ET conditions in the backstepping procedure; 2) the computation of virtual control laws; and 3) the redundant triggering of events when the system states converge to a desired region. The validity of the presented scheme is demonstrated by simulation results.

The Calderón problem for the fractional Schrödinger equation

Abstract: We show global uniqueness in an inverse problem for the fractional Schrodinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where measurements are taken in arbitrary open, possibly disjoint, subsets of the exterior. The results apply in any dimension ≥1 and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically different from the traditional Calderon problem.

A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case

Abstract: A key element of understanding the efficacy of overparameterized neural networks is characterizing how they represent functions as the number of weights in the network approaches infinity. In this paper, we characterize the norm required to realize any function as a single hidden-layer ReLU network with an unbounded number of units (infinite width), but where the Euclidean norm of the weights is bounded, including precisely characterizing which functions can be realized with finite norm. This was settled for univariate functions in Savarese et al. (2019), where it was shown that the required norm is determined by the L1-norm of the second derivative of the function. We extend the characterization to multi-variate functions (i.e., multiple input units), relating the required norm to the L1-norm of the Radon transform of a higher-order Laplacian of the function. This characterization allows us to show that all functions in a Sobolev space, can be represented with bounded norm, to calculate the required norm for several specific functions, and to obtain a depth separation result. These results have important implications for understanding generalization performance and the distinction between neural networks and more traditional kernel learning.

The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: A Typical Case

Abstract: The Quantum Approximate Optimization Algorithm can naturally be applied to combinatorial search problems on graphs. The quantum circuit has p applications of a unitary operator that respects the locality of the graph. On a graph with bounded degree, with p small enough, measurements of distant qubits in the state output by the QAOA give uncorrelated results. We focus on finding big independent sets in random graphs with dn/2 edges keeping d fixed and n large. Using the Overlap Gap Property of almost optimal independent sets in random graphs, and the locality of the QAOA, we are able to show that if p is less than a d-dependent constant times log n, the QAOA cannot do better than finding an independent set of size .854 times the optimal for d large. Because the logarithm is slowly growing, even at one million qubits we can only show that the algorithm is blocked if p is in single digits. At higher p the algorithm "sees" the whole graph and we have no indication that performance is limited.

Decentralized Optimization Over Time-Varying Directed Graphs With Row and Column-Stochastic Matrices

Abstract: In this article, we provide a distributed optimization algorithm, termed as TV- $\mathcal {AB}$ , that minimizes a sum of convex functions over time-varying, random directed graphs. Contrary to the existing work, the algorithm we propose does not require eigenvector estimation to estimate the (non- $\mathbf {1}$ ) Perron eigenvector of a stochastic matrix. Instead, the proposed approach relies on a novel information mixing approach that exploits both row- and column-stochastic weights to achieve agreement toward the optimal solution when the underlying graph is directed. We show that TV- $\mathcal {AB}$ converges linearly to the optimal solution when the global objective is smooth and strongly convex, and the underlying time-varying graphs exhibit bounded connectivity, i.e., a union of every $C$ consecutive graphs is strongly connected. We derive the convergence results based on the stability analysis of a linear system of inequalities along with a matrix perturbation argument. Simulations confirm the findings in this article.

Reasoning About Generalization via Conditional Mutual Information

Abstract: We provide an information-theoretic framework for studying the generalization properties of machine learning algorithms. Our framework ties together existing approaches, including uniform convergence bounds and recent methods for adaptive data analysis. Specifically, we use Conditional Mutual Information (CMI) to quantify how well the input (i.e., the training data) can be recognized given the output (i.e., the trained model) of the learning algorithm. We show that bounds on CMI can be obtained from VC dimension, compression schemes, differential privacy, and other methods. We then show that bounded CMI implies various forms of generalization.

Learning Open Set Network with Discriminative Reciprocal Points

Abstract: Open set recognition is an emerging research area that aims to simultaneously classify samples from predefined classes and identify the rest as ‘unknown’. In this process, one of the key challenges is to reduce the risk of generalizing the inherent characteristics of numerous unknown samples learned from a small amount of known data. In this paper, we propose a new concept, Reciprocal Point, which is the potential representation of the extra-class space corresponding to each known category. The sample can be classified to known or unknown by the otherness with reciprocal points. To tackle the open set problem, we offer a novel open space risk regularization term. Based on the bounded space constructed by reciprocal points, the risk of unknown is reduced through multi-category interaction. The novel learning framework called Reciprocal Point Learning (RPL), which can indirectly introduce the unknown information into the learner with only known classes, so as to learn more compact and discriminative representations. Moreover, we further construct a new large-scale challenging aircraft dataset for open set recognition: Aircraft 300 (Air-300). Extensive experiments on multiple benchmark datasets indicate that our framework is significantly superior to other existing approaches and achieves state-of-the-art performance on standard open set benchmarks.

Tempered Sigmoid Activations for Deep Learning with Differential Privacy.

Abstract: Because learning sometimes involves sensitive data, machine learning algorithms have been extended to offer privacy for training data. In practice, this has been mostly an afterthought, with privacy-preserving models obtained by re-running training with a different optimizer, but using the model architectures that already performed well in a non-privacy-preserving setting. This approach leads to less than ideal privacy/utility tradeoffs, as we show here. Instead, we propose that model architectures are chosen ab initio explicitly for privacy-preserving training. To provide guarantees under the gold standard of differential privacy, one must bound as strictly as possible how individual training points can possibly affect model updates. In this paper, we are the first to observe that the choice of activation function is central to bounding the sensitivity of privacy-preserving deep learning. We demonstrate analytically and experimentally how a general family of bounded activation functions, the tempered sigmoids, consistently outperform unbounded activation functions like ReLU. Using this paradigm, we achieve new state-of-the-art accuracy on MNIST, FashionMNIST, and CIFAR10 without any modification of the learning procedure fundamentals or differential privacy analysis.

Twin-width I: tractable FO model checking

Abstract: Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA '14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, $K_t$-free unit $d$-dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of $d$-contractions, witness that the twin-width is at most $d$. We show that FO model checking, that is deciding if a given first-order formula $\phi$ evaluates to true for a given binary structure $G$ on a domain $D$, is FPT in $|\phi|$ on classes of bounded twin-width, provided the witness is given. More precisely, being given a $d$-contraction sequence for $G$, our algorithm runs in time $f(d,|\phi|) \cdot |D|$ where $f$ is a computable but non-elementary function. We also prove that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS '15].

Why fractional derivatives with nonsingular kernels should not be used

Abstract: In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time $t=0$ is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.

AdS3 solutions in massive IIA with small N$$ \mathcal{N} $$ = (4, 0) supersymmetry

Abstract: We study AdS3 x S 2 solutions in massive IIA that preserve small $$ \mathcal{N} $$ = (4, 0) supersymmetry in terms of an SU(2)-structure on the remaining internal space. We find two new classes of solutions that are warped products of the form AdS3 x S2 x M4 x ℝ. For the first, M4=CY2 and we find a generalisation of a D4-D8 system involving possible additional branes. For the second, M4 need only be Kahler, and we find a generalisation of the T-dual of solutions based on D3-branes wrapping curves in the base of an elliptically fibered Calabi-Yau 3-fold. Within these classes we find many new locally compact solutions that are foliations of AdS3 x S2 x CY2 over an interval, bounded by various D brane and O plane behaviours. We comment on how these local solutions may be used as the building blocks of infinite classes of global solutions glued together with defect branes. Utilising T-duality we find two new classes of AdS3 x S3 x M4 solutions in liB. The first backreacts D5s and KK monopoles on the D1-D5 near horizon. The second is a generalisation of the solutions based on D3-branes wrapping curves in the base of an elliptically fibered CY3 that includes non trivial 3-form flux.

Reinforcement Learning with General Value Function Approximation: Provably Efficient Approach via Bounded Eluder Dimension

Abstract: Value function approximation has demonstrated phenomenal empirical success in reinforcement learning (RL). Nevertheless, despite a handful of recent progress on developing theory for RL with linear function approximation, the understanding of general function approximation schemes largely remains missing. In this paper, we establish a provably efficient RL algorithm with general value function approximation. We show that if the value functions admit an approximation with a function class $\mathcal{F}$, our algorithm achieves a regret bound of $\widetilde{O}(\mathrm{poly}(dH)\sqrt{T})$ where $d$ is a complexity measure of $\mathcal{F}$ that depends on the eluder dimension [Russo and Van Roy, 2013] and log-covering numbers, $H$ is the planning horizon, and $T$ is the number interactions with the environment. Our theory generalizes recent progress on RL with linear value function approximation and does not make explicit assumptions on the model of the environment. Moreover, our algorithm is model-free and provides a framework to justify the effectiveness of algorithms used in practice.

Novel Nonlinear Hypothesis for the Delta Parallel Robot Modeling

Abstract: In previous investigations, the nonlinear hypothesis use the linear bounded maps. Nonlinear hypothesis are described as the combination of the first order terms, and after of the mentioned combination, one bounded map is applied to alter the result. This document proposes two nonlinear hypothesis which use different structures instead of using the linear bounded maps. They are termed as novel nonlinear hypothesis and second order nonlinear hypothesis and their goal is to improve the second order processes modeling. The proposed nonlinear hypothesis are described as the combination of the first order and second order terms. Since the delta parallel robot is a second order process, it is an excellent platform to prove the effectiveness of the two proposed hypothesis.

Extremal Effective Field Theories

Abstract: Effective field theories (EFT) parameterize the long-distance effects of short-distance dynamics whose details may or may not be known. It is known that EFT coefficients must obey certain positivity constraints if causality and unitarity are satisfied at all scales. We explore those constraints from the perspective of 2 to 2 scattering amplitudes of a light real scalar field, using semi-definite programming to carve out the space of allowed EFT coefficients for a given mass threshold M. We point out that all EFT parameters are bounded both below and above, effectively showing that dimensional analysis scaling is a consequence of causality. This includes the coefficients of four- and six-derivative interactions. We present simple extremal amplitudes which realize, or "rule in", kinks in coefficient space and whose convex hull span a large fraction of the allowed space.

Robust Bipartite Consensus and Tracking Control of High-Order Multiagent Systems With Matching Uncertainties and Antagonistic Interactions

Abstract: This paper is concerned with general coopetition networks with signed graphs, based on which both the bipartite consensus and tracking control problems for networked systems subject to nonidentical matching uncertainties are studied. For the case of undirected and connected communication graphs, we propose a distributed discontinuous nonlinear controller which can achieve the bipartite consensus. To cancel the chattering phenomenon of the discontinuous controller, a continuous one is designed by using the boundary layer technique, under which the bipartite consensus error is shown to be uniformly ultimately bounded and can exponentially converge to a small adjustable bounded set. Further, considering the case of a leader having a bounded control action, we present a continuous controller to guarantee the ultimate boundedness of the bipartite tracking error.