Authors:Nan Liu, Boling Guo Pages: 1 - 46 Abstract: Analysis and Applications, Ahead of Print. The large-time behavior of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane is established. Our approach uses the unified transform method of Fokas and the nonlinear steepest descent method of Deift and Zhou. Citation: Analysis and Applications PubDate: 2020-01-23T08:00:00Z DOI: 10.1142/S0219530519500210

Authors:Junping Li, Lan Cheng, Anthony G. Pakes, Anyue Chen, Liuyan Li Pages: 1 - 22 Abstract: Analysis and Applications, Ahead of Print. Large deviation rates are determined for quantities associated with a Markov branching process [math] having offspring mean [math] and split rate [math]. The principal quantities examined are [math] and [math], where [math] is the almost sure limit of an appropriately normed version of [math]. Modifications and conditional versions are examined. Some of this requires determination of the asymptotic behavior of harmonic moments [math]. Citation: Analysis and Applications PubDate: 2020-01-21T08:00:00Z DOI: 10.1142/S0219530519500209

Authors:Baohuai Sheng, Haizhang Zhang Pages: 79 - 108 Abstract: Analysis and Applications, Volume 18, Issue 01, Page 79-108, January 2020. It is known that one aim of semi-supervised learning is to improve the prediction performance using a few labeled data with a large set of unlabeled data. Recently, a Laplacian regularized semi-supervised learning gradient (LapRSSLG) algorithm associated with data adjacency graph edge weights is proposed in the literature. The algorithm receives success in applications, but there is no theory on the performance analysis. In this paper, an explicit learning rate estimate for the algorithm is provided, which shows that the convergence is indeed controlled by the unlabeled data. Citation: Analysis and Applications PubDate: 2019-12-16T08:00:00Z DOI: 10.1142/S0219530519410033

Authors:Ting Hu, Jun Fan, Dao-Hong Xiang Pages: 109 - 127 Abstract: Analysis and Applications, Volume 18, Issue 01, Page 109-127, January 2020. In this paper, we establish the error analysis for distributed pairwise learning with multi-penalty regularization, based on a divide-and-conquer strategy. We demonstrate with [math]-error bound that the learning performance of this distributed learning scheme is as good as that of a single machine which could process the whole data. With semi-supervised data, we can relax the restriction of the number of local machines and enlarge the range of the target function to guarantee the optimal learning rate. As a concrete example, we show that the work in this paper can apply to the distributed pairwise learning algorithm with manifold regularization. Citation: Analysis and Applications PubDate: 2019-12-16T08:00:00Z DOI: 10.1142/S0219530519410045

Authors:Fusheng Lv, Jun Fan Pages: 1 - 18 Abstract: Analysis and Applications, Ahead of Print. Correntropy-based learning has achieved great success in practice during the last decades. It is originated from information-theoretic learning and provides an alternative to classical least squares method in the presence of non-Gaussian noise. In this paper, we investigate the theoretical properties of learning algorithms generated by Tikhonov regularization schemes associated with Gaussian kernels and correntropy loss. By choosing an appropriate scale parameter of Gaussian kernel, we show the polynomial decay of approximation error under a Sobolev smoothness condition. In addition, we employ a tight upper bound for the uniform covering number of Gaussian RKHS in order to improve the estimate of sample error. Based on these two results, we show that the proposed algorithm using varying Gaussian kernel achieves the minimax rate of convergence (up to a logarithmic factor) without knowing the smoothness level of the regression function. Citation: Analysis and Applications PubDate: 2019-11-11T06:55:29Z DOI: 10.1142/S0219530519410124

Authors:Chenxi Chen, Yunmei Chen, Xiaojing Ye Pages: 1 - 25 Abstract: Analysis and Applications, Ahead of Print. We consider a class of convex decentralized consensus optimization problems over connected multi-agent networks. Each agent in the network holds its local objective function privately, and can only communicate with its directly connected agents during the computation to find the minimizer of the sum of all objective functions. We propose a randomized incremental primal-dual method to solve this problem, where the dual variable over the network in each iteration is only updated at a randomly selected node, whereas the dual variables elsewhere remain the same as in the previous iteration. Thus, the communication only occurs in the neighborhood of the selected node in each iteration and hence can greatly reduce the chance of communication delay and failure in the standard fully synchronized consensus algorithms. We provide comprehensive convergence analysis including convergence rates of the primal residual and consensus error of the proposed algorithm, and conduct numerical experiments to show its performance using both uniform sampling and important sampling as node selection strategy. Citation: Analysis and Applications PubDate: 2019-11-07T06:08:24Z DOI: 10.1142/S0219530519410082

Authors:Meipeng Zhi, Yuesheng Xu Pages: 1 - 23 Abstract: Analysis and Applications, Ahead of Print. We develop a numerical method for construction of an adaptive display image from a given display image which is an artificial scene displayed in a computer screen. The adaptive display image is encoded on an adaptive pixel mesh obtained by a merging scheme from the original pixel mesh. The cardinality of the adaptive pixel mesh is significantly less than that of the original pixel mesh. The resulting adaptive display image is the best [math] piecewise constant approximation of the original display image. Under the assumption that a natural image, the real scene that we see, belongs to a Besov space, we provide the optimal [math] error estimate between the adaptive display image and its original natural image. Experimental results are presented to demonstrate the visual quality, the approximation accuracy and the computational complexity of the adaptive display image. Citation: Analysis and Applications PubDate: 2019-11-07T06:08:24Z DOI: 10.1142/S0219530519410112

Authors:Simon Foucart, Rémi Gribonval, Laurent Jacques, Holger Rauhut Pages: 1 - 24 Abstract: Analysis and Applications, Ahead of Print. We investigate the problem of recovering jointly [math]-rank and [math]-bisparse matrices from as few linear measurements as possible, considering arbitrary measurements as well as rank-one measurements. In both cases, we show that [math] measurements make the recovery possible in theory, meaning via a nonpractical algorithm. In case of arbitrary measurements, we investigate the possibility of achieving practical recovery via an iterative-hard-thresholding algorithm when [math] for some exponent [math]. We show that this is feasible for [math], and that the proposed analysis cannot cover the case [math]. The precise value of the optimal exponent [math] is the object of a question, raised but unresolved in this paper, about head projections for the jointly low-rank and bisparse structure. Some related questions are partially answered in passing. For rank-one measurements, we suggest on arcane grounds an iterative-hard-thresholding algorithm modified to exploit the nonstandard restricted isometry property obeyed by this type of measurements. Citation: Analysis and Applications PubDate: 2019-11-07T06:08:23Z DOI: 10.1142/S0219530519410094

Authors:Liren Huang, Chunguang Liu, Lulin Tan, Qi Ye Pages: 1 - 22 Abstract: Analysis and Applications, Ahead of Print. In this paper, we generalize the representer theorems in Banach spaces by the theory of nonsmooth analysis. The generalized representer theorems assure that the regularized learning models can be constructed by the nonconvex loss functions, the generalized training data, and the general Banach spaces which are nonreflexive, nonstrictly convex, and nonsmooth. Specially, the sparse representations of the regularized learning in 1-norm reproducing kernel Banach spaces are shown by the generalized representer theorems. Citation: Analysis and Applications PubDate: 2019-11-07T06:08:23Z DOI: 10.1142/S0219530519410100

Authors:Xianpeng Mao, Gonglin Yuan, Yuning Yang Pages: 1 - 19 Abstract: Analysis and Applications, Ahead of Print. Though the alternating least squares algorithm (ALS), as a classic and easily implemented algorithm, has been widely applied to tensor decomposition and approximation problems, it has some drawbacks: the convergence of ALS is not guaranteed, and the swamp phenomenon appears in some cases, causing the convergence rate to slow down dramatically. To overcome these shortcomings, the regularized-ALS algorithm (RALS) was proposed in the literature. By employing the optimal step-size selection rule, we develop a self-adaptive regularized alternating least squares method (SA-RALS) to accelerate RALS in this paper. Theoretically, we show that the step-size is always larger than unity, and can be larger than [math], which is quite different from several optimization algorithms. Furthermore, under mild assumptions, we prove that the whole sequence generated by SA-RALS converges to a stationary point of the objective function. Numerical results verify that the SA-RALS performs better than RALS in terms of the number of iterations and the CPU time. Citation: Analysis and Applications PubDate: 2019-10-29T07:36:38Z DOI: 10.1142/S0219530519410057

Authors:Saverio Salzo, Johan A. K. Suykens Pages: 1 - 35 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study the variational problem associated to support vector regression in Banach function spaces. Using the Fenchel–Rockafellar duality theory, we give an explicit formulation of the dual problem as well as of the related optimality conditions. Moreover, we provide a new computational framework for solving the problem which relies on a tensor-kernel representation. This analysis overcomes the typical difficulties connected to learning in Banach spaces. We finally present a large class of tensor-kernels to which our theory fully applies: power series tensor kernels. This type of kernels describes Banach spaces of analytic functions and includes generalizations of the exponential and polynomial kernels as well as, in the complex case, generalizations of the Szegö and Bergman kernels. Citation: Analysis and Applications PubDate: 2019-10-29T07:36:36Z DOI: 10.1142/S0219530519410069

Authors:Cheng Wang, Ting Hu Pages: 1 - 30 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study online algorithm for pairwise problems generated from the Tikhonov regularization scheme associated with the least squares loss function and a reproducing kernel Hilbert space (RKHS). This work establishes the convergence for the last iterate of the online pairwise algorithm with the polynomially decaying step sizes and varying regularization parameters. We show that the obtained error rate in [math]-norm can be nearly optimal in the minimax sense under some mild conditions. Our analysis is achieved by a sharp estimate for the norms of the learning sequence and the characterization of RKHS using its associated integral operators and probability inequalities for random variables with values in a Hilbert space. Citation: Analysis and Applications PubDate: 2019-10-29T07:36:35Z DOI: 10.1142/S0219530519410070

Authors:L. Agud, J. M. Calabuig, E. A. Sánchez Pérez Pages: 1 - 25 Abstract: Analysis and Applications, Ahead of Print. Let [math] be a finite measure space and consider a Banach function space [math]. Motivated by some previous papers and current applications, we provide a general framework for representing reproducing kernel Hilbert spaces as subsets of Köthe–Bochner (vector-valued) function spaces. We analyze operator-valued kernels [math] that define integration maps [math] between Köthe–Bochner spaces of Hilbert-valued functions [math] We show a reduction procedure which allows to find a factorization of the corresponding kernel operator through weighted Bochner spaces [math] and [math] — where [math] — under the assumption of [math]-concavity of [math] Equivalently, a new kernel obtained by multiplying [math] by scalar functions can be given in such a way that the kernel operator is defined from [math] to [math] in a natural way. As an application, we prove a new version of Mercer Theorem for matrix-valued weighted kernels. Citation: Analysis and Applications PubDate: 2019-10-01T06:25:18Z DOI: 10.1142/S0219530519500179

Authors:Wei Yan, Yongsheng Li, Jianhua Huang, Jinqiao Duan Pages: 1 - 54 Abstract: Analysis and Applications, Ahead of Print. The goal of this paper is three-fold. First, we prove that the Cauchy problem for a generalized KP-I equation ut + Dx α∂ xu + ∂x−1∂ y2u + 1 2∂x(u2) = 0,α ≥ 4 is locally well-posed in the anisotropic Sobolev spaces [math] with [math] and [math]. Second, we prove that the Cauchy problem is globally well-posed in [math] with [math] if [math]. Finally, we show that the Cauchy problem is globally well-posed in [math] with [math] if [math] Our result improves the result of Saut and Tzvetkov [The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl. 79 (2000) 307–338] and Li and Xiao [Well-posedness of the fifth order Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl. 90 (2008) 338–352]. Citation: Analysis and Applications PubDate: 2019-10-01T06:25:18Z DOI: 10.1142/S0219530519500180

Authors:Bao-Huai Sheng, Jian-Li Wang Pages: 1 - 24 Abstract: Analysis and Applications, Ahead of Print. [math]-functionals are used in learning theory literature to study approximation errors in kernel-based regularization schemes. In this paper, we study the approximation error and [math]-functionals in [math] spaces with [math]. To this end, we give a new viewpoint for a reproducing kernel Hilbert space (RKHS) from a fractional derivative and treat powers of the induced integral operator as fractional derivatives of various orders. Then a generalized translation operator is defined by Fourier multipliers, with which a generalized modulus of smoothness is defined. Some general strong equivalent relations between the moduli of smoothness and the [math]-functionals are established. As applications, some strong equivalent relations between these two families of quantities on the unit sphere and the unit ball are provided explicitly. Citation: Analysis and Applications PubDate: 2019-10-01T06:25:17Z DOI: 10.1142/S0219530519500192

Authors:Xin Zhong Pages: 1 - 29 Abstract: Analysis and Applications, Ahead of Print. We study the Cauchy problem of nonhomogeneous magneto-micropolar fluid system with zero density at infinity in the entire space [math]. We prove that the system admits a unique local strong solution provided the initial density and the initial magnetic field decay not too slowly at infinity. In particular, there is no need to require any Choe–Kim type compatibility condition for the initial data. Citation: Analysis and Applications PubDate: 2019-09-09T08:02:45Z DOI: 10.1142/S0219530519500167

Authors:Lucian Coroianu, Danilo Costarelli, Sorin G. Gal, Gianluca Vinti Pages: 1 - 26 Abstract: Analysis and Applications, Ahead of Print. In a recent paper, for max-product sampling operators based on general kernels with bounded generalized absolute moments, we have obtained several pointwise and uniform convergence properties on bounded intervals or on the whole real axis, including a Jackson-type estimate in terms of the first uniform modulus of continuity. In this paper, first, we prove that for the Kantorovich variants of these max-product sampling operators, under the same assumptions on the kernels, these convergence properties remain valid. Here, we also establish the [math] convergence, and quantitative estimates with respect to the [math] norm, [math]-functionals and [math]-modulus of continuity as well. The results are tested on several examples of kernels and possible extensions to higher dimensions are suggested. Citation: Analysis and Applications PubDate: 2019-08-19T06:31:09Z DOI: 10.1142/S0219530519500155

Authors:Ingo Gühring, Gitta Kutyniok, Philipp Petersen Pages: 1 - 57 Abstract: Analysis and Applications, Ahead of Print. We analyze to what extent deep Rectified Linear Unit (ReLU) neural networks can efficiently approximate Sobolev regular functions if the approximation error is measured with respect to weaker Sobolev norms. In this context, we first establish upper approximation bounds by ReLU neural networks for Sobolev regular functions by explicitly constructing the approximate ReLU neural networks. Then, we establish lower approximation bounds for the same type of function classes. A trade-off between the regularity used in the approximation norm and the complexity of the neural network can be observed in upper and lower bounds. Our results extend recent advances in the approximation theory of ReLU networks to the regime that is most relevant for applications in the numerical analysis of partial differential equations. Citation: Analysis and Applications PubDate: 2019-08-19T06:31:08Z DOI: 10.1142/S0219530519410021

Authors:Palle E. T. Jorgensen, James F. Tian Pages: 1 - 31 Abstract: Analysis and Applications, Ahead of Print. We study reduction schemes for functions of “many” variables into system of functions in one variable. Our setting includes infinite dimensions. Following Cybenko–Kolmogorov, the outline for our results is as follows: We present explicit reduction schemes for multivariable problems, covering both a finite, and an infinite, number of variables. Starting with functions in “many” variables, we offer constructive reductions into superposition, with component terms, that make use of only functions in one variable, and specified choices of coordinate directions. Our proofs are transform based, using explicit transforms, Fourier and Radon; as well as multivariable Shannon interpolation. Citation: Analysis and Applications PubDate: 2019-08-05T06:44:19Z DOI: 10.1142/S021953051941001X

Authors:Ahmed Abdeljawad, Sandro Coriasco, Joachim Toft Pages: 1 - 61 Abstract: Analysis and Applications, Ahead of Print. We deduce one-parameter group properties for pseudo-differential operators [math], where [math] belongs to the class [math] of certain Gevrey symbols. We use this to show that there are pseudo-differential operators [math] and [math] which are inverses to each other, where [math] and [math]. We apply these results to deduce lifting property for modulation spaces and construct explicit isomorphisms between them. For each weight functions [math] moderated by GRS submultiplicative weights, we prove that the Toeplitz operator (or localization operator) [math] is an isomorphism from [math] to [math] for every [math]. Citation: Analysis and Applications PubDate: 2019-07-30T02:40:53Z DOI: 10.1142/S0219530519500143

Authors:Wei Shen, Zhenhuan Yang, Yiming Ying, Xiaoming Yuan Pages: 1 - 41 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study the stability and its trade-off with optimization error for stochastic gradient descent (SGD) algorithms in the pairwise learning setting. Pairwise learning refers to a learning task which involves a loss function depending on pairs of instances among which notable examples are bipartite ranking, metric learning, area under ROC curve (AUC) maximization and minimum error entropy (MEE) principle. Our contribution is twofolded. Firstly, we establish the stability results for SGD for pairwise learning in the convex, strongly convex and non-convex settings, from which generalization errors can be naturally derived. Secondly, we establish the trade-off between stability and optimization error of SGD algorithms for pairwise learning. This is achieved by lower-bounding the sum of stability and optimization error by the minimax statistical error over a prescribed class of pairwise loss functions. From this fundamental trade-off, we obtain lower bounds for the optimization error of SGD algorithms and the excess expected risk over a class of pairwise losses. In addition, we illustrate our stability results by giving some specific examples of AUC maximization, metric learning and MEE. Citation: Analysis and Applications PubDate: 2019-07-24T03:00:36Z DOI: 10.1142/S0219530519400062

Authors:Huiping Li, Song Li, Yu Xia Pages: 1 - 26 Abstract: Analysis and Applications, Ahead of Print. In this paper, we consider the noisy phase retrieval problem which occurs in many different areas of science and physics. The PhaseMax algorithm is an efficient convex method to tackle with phase retrieval problem. On the basis of this algorithm, we propose two kinds of extended formulations of the PhaseMax algorithm, namely, PhaseMax with bounded and non-negative noise and PhaseMax with outliers to deal with the phase retrieval problem under different noise corruptions. Then we prove that these extended algorithms can stably recover real signals from independent sub-Gaussian measurements under optimal sample complexity. Specially, such results remain valid in noiseless case. As we can see, these results guarantee that a broad range of random measurements such as Bernoulli measurements with erasures can be applied to reconstruct the original signals by these extended PhaseMax algorithms. Finally, we demonstrate the effectiveness of our extended PhaseMax algorithm through numerical simulations. We find that with the same initialization, extended PhaseMax algorithm outperforms Truncated Wirtinger Flow method, and recovers the signal with corrupted measurements robustly. Citation: Analysis and Applications PubDate: 2019-07-24T03:00:33Z DOI: 10.1142/S0219530519400049

Authors:Michel Chipot, Jérôme Droniou, Gabriela Planas, James C. Robinson, Wei Xue Pages: 1 - 25 Abstract: Analysis and Applications, Ahead of Print. We treat three problems on a two-dimensional “punctured periodic domain”: we take [math], where [math] and [math] is the closure of an open connected set that is star-shaped with respect to [math] and has a [math] boundary. We impose periodic boundary conditions on the boundary of [math], and Dirichlet boundary conditions on [math]. In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier–Stokes equations, all with a fixed forcing function [math], and examine the behavior of solutions as [math]. In all three cases we show convergence of the solutions to those of the limiting problem, i.e. the problem posed on all of [math] with periodic boundary conditions. Citation: Analysis and Applications PubDate: 2019-07-05T08:54:22Z DOI: 10.1142/S0219530519500118

Authors:Ben Duan, Zhen Luo, Yan Zhou Pages: 1 - 26 Abstract: Analysis and Applications, Ahead of Print. In this paper, we consider the Cauchy problem of a viscous compressible shallow water equations with the Coriolis force term and non-constant viscosities. More precisely, the viscous coefficients are constants multiple of height, the equations are degenerate when vacuum appears. For initial data allowing vacuum, the local existence of strong solution is obtained and a blow-up criterion is established. Citation: Analysis and Applications PubDate: 2019-07-05T08:54:22Z DOI: 10.1142/S021953051950012X

Authors:Mourad E. H. Ismail Pages: 1 - 26 Abstract: Analysis and Applications, Ahead of Print. We study the moment problem associated with the Al-Salam–Chihara polynomials in some detail providing raising (creation) and lowering (annihilation) operators, Rodrigues formula, and a second-order operator equation involving the Askey–Wilson operator. A new infinite family of weight functions is also given. Sufficient conditions for functions to be weight functions for the [math]-Hermite, [math]-Laguerre and Stieltjes–Wigert polynomials are established and used to give new infinite families of absolutely continuous orthogonality measures for each of these polynomials. Citation: Analysis and Applications PubDate: 2019-06-14T06:00:54Z DOI: 10.1142/S0219530519500088

Authors:Ana F. Loureiro, Walter Van Assche Pages: 1 - 62 Abstract: Analysis and Applications, Ahead of Print. We characterize all the multiple orthogonal three-fold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as [math]-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a [math]-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them [math]-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni. Citation: Analysis and Applications PubDate: 2019-06-14T06:00:53Z DOI: 10.1142/S0219530519500106

Authors:Yu-Tian Li, Xiang-Sheng Wang, Roderick Wong Pages: 1 - 34 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study the asymptotic behavior of the Wilson polynomials [math] as their degree tends to infinity. These polynomials lie on the top level of the Askey scheme of hypergeometric orthogonal polynomials. Infinite asymptotic expansions are derived for these polynomials in various cases, for instance, (i) when the variable [math] is fixed and (ii) when the variable is rescaled as [math] with [math]. Case (ii) has two subcases, namely, (a) zero-free zone ([math]) and (b) oscillatory region [math]. Corresponding results are also obtained in these cases (iii) when [math] lies in a neighborhood of the transition point [math], and (iv) when [math] is in the neighborhood of the transition point [math]. The expansions in the last two cases hold uniformly in [math]. Case (iv) is also the only unsettled case in a sequence of works on the asymptotic analysis of linear difference equations. Citation: Analysis and Applications PubDate: 2019-05-16T02:23:12Z DOI: 10.1142/S0219530519500076

Authors:Elena Cordero, S. Ivan Trapasso Pages: 1 - 38 Abstract: Analysis and Applications, Ahead of Print. The Wigner distribution is a milestone of Time–frequency Analysis. In order to cope with its drawbacks while preserving the desirable features that made it so popular, several kinds of modifications have been proposed. This contribution fits into this perspective. We introduce a family of phase-space representations of Wigner type associated with invertible matrices and explore their general properties. As a main result, we provide a characterization for the Cohen’s class [L. Cohen, Generalized phase-space distribution functions, J. Math. Phys. 7 (1996) 781–786; Time–frequency Analysis (Prentice Hall, New Jersey, 1995)]. This feature suggests to interpret this family of representations as linear perturbations of the Wigner distribution. We show which of its properties survive under linear perturbations and which ones are truly distinctive of its central role. Citation: Analysis and Applications PubDate: 2019-03-25T02:23:53Z DOI: 10.1142/S0219530519500052