Abstract: In this paper, we present the Lipschitz regularization theory and algorithms for a novel Loss-Sensitive Generative Adversarial Network (LS-GAN). Specifically, it trains a loss function to distinguish between real and fake samples by designated margins, while learning a generator alternately to produce realistic samples by minimizing their losses. The LS-GAN further regularizes its loss function with a Lipschitz regularity condition on the density of real data, yielding a regularized model that can better generalize to produce new data from a reasonable number of training examples than the classic GAN. We will further present a Generalized LS-GAN (GLS-GAN) and show it contains a large family of regularized GAN models, including both LS-GAN and Wasserstein GAN, as its special cases. Compared with the other GAN models, we will conduct experiments to show both LS-GAN and GLS-GAN exhibit competitive ability in generating new images in terms of the Minimum Reconstruction Error (MRE) assessed on a separate test set. We further extend the LS-GAN to a conditional form for supervised and semi-supervised learning problems, and demonstrate its outstanding performance on image classification tasks.

Abstract: We propose a sequential minimal optimization method for quantum-classical hybrid algorithms, which converges faster, is robust against statistical error, and is hyperparameter-free. Specifically, the optimization problem of the parameterized quantum circuits is divided into solvable subproblems by considering only a subset of the parameters. In fact, if we choose a single parameter, the cost function becomes a simple sine curve with period $2\pi$, and hence we can exactly minimize with respect to the chosen parameter. Furthermore, even in general cases, the cost function is given by a simple sum of trigonometric functions with certain periods and hence can be minimized by using a classical computer. By repeatedly performing this procedure, we can optimize the parameterized quantum circuits so that the cost function becomes as small as possible. We perform numerical simulations and compare the proposed method with existing gradient-free and gradient-based optimization algorithms. We find that the proposed method substantially outperforms the existing optimization algorithms and converges to a solution almost independent of the initial choice of the parameters. This accelerates almost all quantum-classical hybrid algorithms readily and would be a key tool for harnessing near-term quantum devices.

Abstract: The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f ( t ) , by a fractional integral operator applied to the derivative f ′ ( t ) . We define a new fractional operator by substituting for this f ′ ( t ) a more general proportional derivative. This new operator can also be written as a Riemann–Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann–Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function.

Abstract: This paper investigates the problem of event-triggered adaptive control for a class of nonlinear systems subject to asymmetric input saturation and time-varying partial state constraints. To facilitate analyzing the influence of asymmetric input saturation, the saturation function is converted into a linear form with respect to control input. To achieve the objective that partial states do not exceed the constraints, a more general form of Lyapunov function is offered. Different from some existing results about output/full state constraints, the proposed scheme only requires that the partial states satisfy the time-varying constraints. Moreover, an event-triggered scheme with a varying threshold is designed to reduce the communication burden. With the time-varying asymmetric barrier Lyapunov functions, a novel event-triggered control scheme is developed, which ensures that partial states are without violation of required constraints and the tracking error converges to a small neighborhood of the origin despite appearing as saturated phenomenon. Eventually, the theoretic results are confirmed by two examples.

Abstract: Neural networks (NNs) are the method of choice for building learning algorithms. They are now being investigated for other numerical tasks such as solving high-dimensional partial differential equations. Their popularity stems from their empirical success on several challenging learning problems (computer chess/Go, autonomous navigation, face recognition). However, most scholars agree that a convincing theoretical explanation for this success is still lacking. Since these applications revolve around approximating an unknown function from data observations, part of the answer must involve the ability of NNs to produce accurate approximations. This article surveys the known approximation properties of the outputs of NNs with the aim of uncovering the properties that are not present in the more traditional methods of approximation used in numerical analysis, such as approximations using polynomials, wavelets, rational functions and splines. Comparisons are made with traditional approximation methods from the viewpoint of rate distortion, i.e. error versus the number of parameters used to create the approximant. Another major component in the analysis of numerical approximation is the computational time needed to construct the approximation, and this in turn is intimately connected with the stability of the approximation algorithm. So the stability of numerical approximation using NNs is a large part of the analysis put forward. The survey, for the most part, is concerned with NNs using the popular ReLU activation function. In this case the outputs of the NNs are piecewise linear functions on rather complicated partitions of the domain of f into cells that are convex polytopes. When the architecture of the NN is fixed and the parameters are allowed to vary, the set of output functions of the NN is a parametrized nonlinear manifold. It is shown that this manifold has certain space-filling properties leading to an increased ability to approximate (better rate distortion) but at the expense of numerical stability. The space filling creates the challenge to the numerical method of finding best or good parameter choices when trying to approximate.

Abstract: Studying degenerate versions of various special polynomials has became an active area of research and has yielded many interesting arithmetic and combinatorial results. Here we introduce a degenerate version of the polylogarithm function, the so-called degenerate polylogarithm function. Then we construct a new type of degenerate Bernoulli polynomial and number, the so-called degenerate poly-Bernoulli polynomial and number, by using the degenerate polylogarithm function, and derive several properties concerning the degenerate poly-Bernoulli numbers.

Abstract: A brief proof of the statement that the zero-set of a nontrivial real-analytic function in $d$-dimensional space has zero measure is provided.

Abstract: A machine is taught by finding the minimum value of the cost function which is induced by learning data. Unfortunately, as the amount of learning increases, the non-liner activation function in the artificial neural network (ANN), the complexity of the artificial intelligence structures, and the cost function’s non-convex complexity all increase. We know that a non-convex function has local minimums, and that the first derivative of the cost function is zero at a local minimum. Therefore, the methods based on a gradient descent optimization do not undergo further change when they fall to a local minimum because they are based on the first derivative of the cost function. This paper introduces a novel optimization method to make machine learning more efficient. In other words, we construct an effective optimization method for non-convex cost function. The proposed method solves the problem of falling into a local minimum by adding the cost function in the parameter update rule of the ADAM method. We prove the convergence of the sequences generated from the proposed method and the superiority of the proposed method by numerical comparison with gradient descent (GD, ADAM, and AdaMax).

Abstract: Theory of analytic functions is one of major fields of modern mathematics. Its application covers broad range of topics of natural science. A complex function f (z), or a function that takes a complex number z as a variable, has various properties that often differ from those of functions that take a real number x as a variable. In particular, the analytic functions hold a paramount position in the complex analysis. In this chapter we explore various features of the analytic functions accordingly. From a practical point of view, the theory of analytic functions is very frequently utilized for the calculation of real definite integrals. For this reason, we describe the related topics together with tangible examples.

Abstract: We consider optimization problems over the Stiefel manifold whose objective function is the summation of a smooth function and a nonsmooth function. Existing methods for solving this kind of proble...

Abstract: In the last past year researchers have relied on the ability of Laplace transform to solve partial, ordinary linear equations with great success. Important analysis in signal analysis including the transfer function, Bode diagram, Nyquist plot and Nichols plot are obtained based on the Laplace transform. The output of the analysis depends only on the results obtained from Laplace transform. However, one weakness of Laplace transform is that the Laplace transform of even function is odd while the Laplace transform of an old function is even which is lack of conservation of properties. On the other hand there exist a similar integral transform known as Sumudu transform has the ability to conserve the properties of the function from real space to complex space. The question that arises in the work, is the following: Can we apply the Sumudu transform to construct new transfer functions that will lead to new Bode, Nichols and Nyquist plots? this question is answered in this work.

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.

Abstract: We analyze local operator growth in nonintegrable quantum many-body systems. A recently introduced universal operator growth hypothesis proposes that the maximal growth of Lanczos coefficients in the continued fraction expansion of the Green's function reflects chaos of the underlying system. We first show that the continued fraction expansion, and the recursion method in general, should be understood in the context of a completely integrable classical dynamics in Krylov space. In particular, the time-correlation function of a physical observable analytically continued to imaginary time is a tau-function of integrable Toda hierarchy. We use this relation to generalize the universal operator growth hypothesis to include arbitrarily ordered correlation functions. We then proceed to analyze the singularity of the time-correlation function, which is an equivalent sign of chaos to the maximal growth of Lanczos coefficients, and we show that it is due to delocalization in Krylov space. We illustrate the general relation between chaos and delocalization using an explicit example of the Sachdev-Ye-Kietaev model.

Abstract: The complex-value-based generalized Dempster–Shafer evidence theory, also called complex evidence theory is a useful methodology to handle uncertainty problems of decision-making on the framework of complex plane. In this paper, we propose a new concept of belief function in complex evidence theory. Furthermore, we analyze the axioms of the proposed belief function, then define a plausibility function in complex evidence theory. The newly defined belief and plausibility functions are the generalizations of the traditional ones in Dempster–Shafer (DS) evidence theory, respectively. In particular, when the complex basic belief assignments are degenerated from complex numbers to classical basic belief assignments (BBAs), the generalized belief and plausibility functions in complex evidence theory degenerate into the traditional belief and plausibility functions in DS evidence theory, respectively. Some special types of the generalized belief function are further discussed as well as their characteristics. In addition, an interval constructed by the generalized belief and plausibility functions can be utilized for fuzzy measure, which provides a promising way to express and model the uncertainty in decision theory.

Abstract: Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integral equations with sparse, noisy, unstructured, and multi-fidelity data. PINNs incorporate all available information into a loss function, thus recasting the original problem into an optimization problem. In this paper, we extend PINNs to parameter and function inference for integral equations such as nonlocal Poisson and nonlocal turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The contribution of the paper is three-fold. First, we propose a unified nonlocal operator, which converges to the classical Laplacian as one of the operator parameters, the nonlocal interaction radius $\delta$ goes to zero, and to the fractional Laplacian as $\delta$ goes to infinity. This universal operator forms a super-set of classical Laplacian and fractional Laplacian operators and, thus, has the potential to fit a broad spectrum of data sets. We provide theoretical convergence rates with respect to $\delta$ and verify them via numerical experiments. Second, we use nPINNs to estimate the two parameters, $\delta$ and $\alpha$. The strong non-convexity of the loss function yielding multiple (good) local minima reveals the occurrence of the operator mimicking phenomenon: different pairs of estimated parameters could produce multiple solutions of comparable accuracy. Third, we propose another nonlocal operator with spatially variable order $\alpha(y)$, which is more suitable for modeling turbulent Couette flow. Our results show that nPINNs can jointly infer this function as well as $\delta$. Also, these parameters exhibit a universal behavior with respect to the Reynolds number, a finding that contributes to our understanding of nonlocal interactions in wall-bounded turbulence.

Abstract: The problem whether a given pair of functions can be used as the kernels of a generalized fractional derivative and the associated generalized fractional integral is reduced to the problem of existence of a solution to the Sonine equation. It is shown for some selected classes of functions that a necessary condition for a function to be the kernel of a fractional derivative is an integrable singularity at 0. It is shown that locally integrable completely monotone functions satisfy the Sonine equation if and only if they are singular at 0.

Abstract: This paper investigates the finite-time tracking control problem of the hypersonic flight vehicle (HFV) with state constraints. Firstly, a control-oriented model is introduced to enable the application of adaptive backstepping scheme. To meet strict requirements in terms of working conditions of HFV, barrier Lyapunov function is adopted to constrain the tracking errors, while piecewise saturation function is constructed to restrict the virtual signals. To guarantee the finite-time convergent property of HFV dynamics, an adaptive scheme in accordance with finite-time stability theory is designed. Meanwhile, a sliding mode differentiator is employed to estimate the derivatives of the virtual control laws. Novel auxiliary systems are then designed to consider the side effects of the possible saturation and to maintain the finite-time convergent property. In the final stage, the effectiveness and performance of the proposed method is demonstrated by numerical simulations.

Abstract: Inverse problems are encountered in many domains of physics, with analytic continuation of the imaginary Green's function into the real frequency domain being a particularly important example. However, the analytic continuation problem is ill defined and currently no analytic transformation for solving it is known. We present a general framework for building an artificial neural network (ANN) that solves this task with a supervised learning approach. Application of the ANN approach to quantum Monte Carlo calculations and simulated Green's function data demonstrates its high accuracy. By comparing with the commonly used maximum entropy approach, we show that our method can reach the same level of accuracy for low-noise input data, while performing significantly better when the noise strength increases. The computational cost of the proposed neural network approach is reduced by almost three orders of magnitude compared to the maximum entropy method.

Abstract: In this paper we define a new (output) multiplicative differential, and the corresponding c -differential uniformity. With this new concept, even for characteristic 2, there are perfect ${c}$ -nonlinear (PcN) functions. We first characterize the ${c}$ -differential uniformity of a function in terms of its Walsh transform. We further look at some of the known perfect nonlinear (PN) functions and show that only one remains a PcN function, under a different condition on the parameters. In fact, the p -ary Gold PN function increases its c -differential uniformity significantly, under some conditions on the parameters. We then precisely characterize the c -differential uniformity of the inverse function (in any dimension and characteristic), relevant for the Rijndael (and Advanced Encryption Standard) block cipher.

Abstract: We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan's constant and $\pi$.

Abstract: In this paper, we focus on the constrained sparse regression problem, where the loss function is convex but nonsmooth and the penalty term is defined by the cardinality function. First, we give an ...

Abstract: We propose two numerical algorithms in the fully nonconvex setting for the minimization of the sum of a smooth function and the composition of a nonsmooth function with a linear operator. The itera...

Abstract: Supersymmetry allows a D4R4 interaction in M-theory, but such an interaction is inconsistent with string theory dualities and so is known to be absent. We provide a novel proof of the absence of the D4R4 M-theory interaction by calculating 4-point scattering amplitudes of 11d supergravitons from ABJM theory. This calculation extends a previous calculation performed to the order corresponding to the R4 interaction. The new ingre- dient in this extension is the interpretation of the fourth derivative of the mass deformed S3 partition function of ABJM theory, which can be determined using supersymmetric localization, as a constraint on the Mellin amplitude associated with the stress tensor mul- tiplet 4-point function. As part of this computation, we relate the 4-point function of the superconformal primary of the stress tensor multiplet of any 3d $$ \mathcal{N} $$ = 8 SCFT to some of the 4-point functions of its superconformal descendants. We also provide a concise formula for a general integrated 4-point function on Sd for any d.

Abstract: In the article, we prove that the function x → (1-x)pK(√x) is logarithmically concave on (0,1) if and only if p ≥ 7/32, the function x → K(√x)/log(1+4/√1-x) is convex on (0,1) and the function x → d2/dx2 [K(√x)- log (1+4/√1-x) is absolutely monotonic on (0,1), where K(x) = ∫π/20 (1-x2 sin2t)-1/2 dt (0 < x < 1) is the complete elliptic integral of the first kind.