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Authors:Razvan C. Fetecau, Hansol Park, Francesco S. Patacchini Pages: 965 - 1017 Abstract: Analysis and Applications, Volume 19, Issue 06, Page 965-1017, November 2021. We investigate a model for collective behavior with intrinsic interactions on Riemannian manifolds. We establish the well-posedness of measure-valued solutions (defined via mass transport) on sphere, as well as investigate the mean-field particle approximation. We study the long-time behavior of solutions to the model on sphere, where the primary goal is to establish sufficient conditions for a consensus state to form asymptotically. Well-posedness of solutions and the formation of consensus are also investigated for other manifolds (e.g., a hypercylinder). Citation: Analysis and Applications PubDate: 2021-04-19T07:00:00Z DOI: 10.1142/S0219530521500081 Issue No:Vol. 19, No. 06 (2021)
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Authors:Ning Bi, Jun Tan, Wai-Shing Tang Pages: 1019 - 1031 Abstract: Analysis and Applications, Volume 19, Issue 06, Page 1019-1031, November 2021. In this paper, we provide a necessary condition and a sufficient condition such that any [math]-sparse vector [math] can be recovered from [math] via [math] local minimization. Moreover, we further verify that the sufficient condition is naturally valid when the restricted isometry constant of the measurement matrix [math] satisfies [math]. Compared with the existing [math] local recoverability condition [math], this result shows that [math] local recoverability contains more measurement matrices. Citation: Analysis and Applications PubDate: 2021-04-10T07:00:00Z DOI: 10.1142/S0219530521500068 Issue No:Vol. 19, No. 06 (2021)
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Authors:Xingcai Zhou, Shaogao Lv Pages: 1033 - 1057 Abstract: Analysis and Applications, Volume 19, Issue 06, Page 1033-1057, November 2021. This paper considers a class of robust estimation problems for varying coefficient dynamic models via wavelet techniques, which can adapt to local features of the underlying functions and has less restriction to the smoothness of the functions. The convergence rates and asymptotic distributions of the robust wavelet-based estimator are established when the design variables are stationary short-range dependent (SRD) and the errors are long-range dependent (LRD). Particularly, a rate of convergence [math] in terms of estimation consistency can be achievable when the true components satisfy certain smoothness for a LRD process. Furthermore, an asymptotic property of the proposed estimator is given to indicate the confidence level of our proposed method for varying coefficient models with LRD. Citation: Analysis and Applications PubDate: 2021-03-06T08:00:00Z DOI: 10.1142/S0219530521500032 Issue No:Vol. 19, No. 06 (2021)
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Authors:Zongmin Wu, Ran Zhang Pages: 1093 - 1111 Abstract: Analysis and Applications, Volume 19, Issue 06, Page 1093-1111, November 2021. The nonlinear chaotic differential/algebraic equation (DAE) has been established to simulate the nonuniform oscillations of the motion of a falling sphere in the non-Newtonian fluid. The DAE is obtained only by learning the experimental data with sparse optimization method. However, the deterministic solution will become increasingly inaccurate for long time approximation of the continuous system. In this paper, we introduce two probabilistic solutions to compute the totally DAE, the Random branch selection iteration (RBSI) and Random switching iteration (RSI). The samples are also taken as the reference trajectory to learn random parameter. The proposed probabilistic solutions can be regarded as the discrete analogues of differential inclusion and switching DAEs, respectively. They have been also compared with the deterministic method, i.e. backward differentiation formula (BDF). The deterministic methods only give limited candidates of all the probability solutions, while the RSI can include all the possible trajectories. The numerical results and statistical information criterion show that RSI can successfully reveal the sustaining instabilities of the motion itself and long time chaotic behavior. Citation: Analysis and Applications PubDate: 2021-05-31T07:00:00Z DOI: 10.1142/S021953052150010X Issue No:Vol. 19, No. 06 (2021)
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Pages: 1113 - 1116 Abstract: Analysis and Applications, Volume 19, Issue 06, Page 1113-1116, November 2021.
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Authors:Yeonjong Shin Pages: 1 - 47 Abstract: Analysis and Applications, Ahead of Print. Deep neural networks have been used in various machine learning applications and achieved tremendous empirical successes. However, training deep neural networks is a challenging task. Many alternatives have been proposed in place of end-to-end back-propagation. Layer-wise training is one of them, which trains a single layer at a time, rather than trains the whole layers simultaneously. In this paper, we study a layer-wise training using a block coordinate gradient descent (BCGD) for deep linear networks. We establish a general convergence analysis of BCGD and found the optimal learning rate, which results in the fastest decrease in the loss. We identify the effects of depth, width, and initialization. When the orthogonal-like initialization is employed, we show that the width of intermediate layers plays no role in gradient-based training beyond a certain threshold. Besides, we found that the use of deep networks could drastically accelerate convergence when it is compared to those of a depth 1 network, even when the computational cost is considered. Numerical examples are provided to justify our theoretical findings and demonstrate the performance of layer-wise training by BCGD. Citation: Analysis and Applications PubDate: 2021-12-31T08:00:00Z DOI: 10.1142/S0219530521500263
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Authors:Wei Li, Jun Xian Pages: 1 - 20 Abstract: Analysis and Applications, Ahead of Print. The set of sampling and reconstruction in trigonometric polynomial spaces will play an important role in signal processing. However, in many applications, the frequencies in trigonometric polynomial spaces are not all integers. In this paper, we consider the problem of weighted random sampling and reconstruction of functions in general multivariate trigonometric polynomial spaces. The sampling set is randomly selected on a bounded cube with a probability distribution. We obtain that with overwhelming probability, the sampling inequality holds and the explicit reconstruction formula succeeds for all functions in the general multivariate trigonometric polynomial spaces when the sampling size is sufficiently large. Citation: Analysis and Applications PubDate: 2021-12-31T08:00:00Z DOI: 10.1142/S0219530521500330
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Authors:Jochen Schmid Pages: 1 - 39 Abstract: Analysis and Applications, Ahead of Print. We deal with monotonic regression of multivariate functions [math] on a compact rectangular domain [math] in [math], where monotonicity is understood in a generalized sense: as isotonicity in some coordinate directions and antitonicity in some other coordinate directions. As usual, the monotonic regression of a given function [math] is the monotonic function [math] that has the smallest (weighted) mean-squared distance from [math]. We establish a simple general approach to compute monotonic regression functions: namely, we show that the monotonic regression [math] of a given function [math] can be approximated arbitrarily well — with simple bounds on the approximation error in both the [math]-norm and the [math]-norm — by the monotonic regression [math] of grid-constant functions [math]. monotonic regression algorithms. We also establish the continuity of the monotonic regression [math] of a continuous function [math] along with an explicit averaging formula for [math]. And finally, we deal with generalized monotonic regression where the mean-squared distance from standard monotonic regression is replaced by more complex distance measures which arise, for instance, in maximum smoothed likelihood estimation. We will see that the solution of such generalized monotonic regression problems is simply given by the standard monotonic regression [math]. Citation: Analysis and Applications PubDate: 2021-12-23T08:00:00Z DOI: 10.1142/S0219530521500299
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Authors:Yulan Wang, Michael Winkler, Zhaoyin Xiang Pages: 1 - 30 Abstract: Analysis and Applications, Ahead of Print. The chemotaxis-Stokes system nt + u ⋅∇n = ∇⋅ (nm−1∇n) −∇⋅ (n∇c),ct + u ⋅∇c = Δc − nc,ut = Δu + ∇P + n∇ϕ,∇⋅ u = 0 is considered subject to the boundary condition (nm−1∇n − n∇c) ⋅ ν = 0,c = c ∗(x,t),u = 0,x ∈ ∂Ω,t> 0, with [math] and a given nonnegative function [math]. In contrast to the well-studied case when the second requirement herein is replaced by a homogeneous Neumann boundary condition for [math], the Dirichlet condition imposed here seems to destroy a natural energy-like property that has formed a core ingredient in the literature by providing comprehensive regularity features of the latter problem. This paper attempts to suitably cope with accordingly poor regularity information in order to nevertheless derive a statement on global existence within a generalized framework of solvability which involves appropriately mild requirements on regularity, but which maintains mass conservation in the first component as a key solution property. Citation: Analysis and Applications PubDate: 2021-12-15T08:00:00Z DOI: 10.1142/S0219530521500275
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Authors:Shengda Zeng, Stanisław Migórski, Domingo A. Tarzia Pages: 1 - 20 Abstract: Analysis and Applications, Ahead of Print. The goal of this paper is to investigate a new class of elliptic mixed boundary value problems involving a nonlinear and nonhomogeneous partial differential operator [math]-Laplacian, and a multivalued term represented by Clarke’s generalized gradient. First, we apply a surjectivity result for multivalued pseudomonotone operators to examine the existence of weak solutions under mild hypotheses. Then, a comparison theorem is delivered, and a convergence result, which reveals the asymptotic behavior of solution when the parameter (heat transfer coefficient) tends to infinity, is obtained. Finally, we establish a continuous dependence result of solution to the boundary value problem on the data. Citation: Analysis and Applications PubDate: 2021-12-09T08:00:00Z DOI: 10.1142/S0219530521500287
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Authors:Weichao Guo, Jianxun He, Huoxiong Wu, Dongyong Yang Pages: 1 - 37 Abstract: Analysis and Applications, Ahead of Print. Let [math], [math] and [math] be a singular or fractional integral operator with homogeneous kernel [math]. In this paper, a CMO type space [math] is introduced and studied. In particular, the relationship between [math] and the Lipchitz space [math] is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator [math] on weighted Lebesgue spaces via functions in [math], and an equivalent characterization of the compactness for [math] via functions in [math] are obtained. Some results are new even in the unweighted setting for the first-order commutators. Citation: Analysis and Applications PubDate: 2021-11-27T08:00:00Z DOI: 10.1142/S0219530521500226
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Authors:Shousheng Luo, Xue-Cheng Tai, Yang Wang Pages: 1 - 17 Abstract: Analysis and Applications, Ahead of Print. We present a novel and computable characterization method for convex shapes. We prove that the shape convexity is equivalent to a quadratic constraint on the associated indicator function. Such a simple characterization method allows us to design efficient algorithms for various applications with convex shape prior. In order to show the effectiveness of the proposed approach, this method is incorporated with a probability-based model to extract an object with convexity prior. The Lagrange multiplier method is used to solve the proposed model. Numerical results on various images show the superiority of the proposed method. Citation: Analysis and Applications PubDate: 2021-11-25T08:00:00Z DOI: 10.1142/S0219530521500238
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Authors:Wei Qu, Charles K. Chui, Guan-Tie Deng, Tao Qian Pages: 1 - 23 Abstract: Analysis and Applications, Ahead of Print. The Dirac-[math] distribution may be realized through sequences of convolutions, the latter being also regarded as approximation to the identity. This paper proposes the real form of pre-orthogonal adaptive Fourier decomposition (POAFD) method to realize fast approximation to the identity. It belongs to sparse representation of signals having potential applications in signal and image analysis. Citation: Analysis and Applications PubDate: 2021-11-24T08:00:00Z DOI: 10.1142/S0219530521500251
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Authors:Guangqing Wang, Wenyi Chen, Jie Yang Pages: 1 - 15 Abstract: Analysis and Applications, Ahead of Print. Let [math] be a Fourier integral operator defined with [math] and [math] satisfying the strong non-degenerate condition. It is shown that [math] is a bounded operator from [math] to [math], if [math], [math], [math] and m ≤ ϱ − n 2 +min 0, n 2 (ϱ − δ) . Citation: Analysis and Applications PubDate: 2021-11-17T08:00:00Z DOI: 10.1142/S0219530521500214
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Authors:Boris Rubin Pages: 1 - 15 Abstract: Analysis and Applications, Ahead of Print. We study the spherical slice transform [math] which assigns to a function [math] on the unit sphere [math] in [math] the integrals of [math] over cross-sections of [math] by [math]-dimensional affine planes passing through the north pole [math]. These transforms are known when [math]. We consider all [math] and obtain an explicit formula connecting [math] with the classical [math]-plane Radon–John transform [math] on [math]. Using this connection, known facts for [math], like inversion formulas, support theorems, representation on zonal functions, and some others, are reformulated for [math]. Citation: Analysis and Applications PubDate: 2021-11-17T08:00:00Z DOI: 10.1142/S021953052150024X
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Authors:Yi-Long Luo, Yangjun Ma Pages: 1 - 64 Abstract: Analysis and Applications, Ahead of Print. The Qian–Sheng model is a system describing the hydrodynamics of nematic liquid crystals in the Q-tensor framework. When the inertial effect is included, it is a hyperbolic-type system involving a second-order material derivative coupling with forced incompressible Navier–Stokes equations. If formally letting the inertial constant [math] go to zero, the resulting system is the corresponding parabolic model. We provide the result on the rigorous justification of this limit in [math] with small initial data, which validates mathematically the parabolic Qian–Sheng model. To achieve this, an initial layer is introduced to not only overcome the disparity of the initial conditions between the hyperbolic and parabolic models, but also make the convergence rate optimal. Moreover, a novel [math]-dependent energy norm is carefully designed, which is non-negative only when [math] is small enough, and handles the difficulty brought by the second-order material derivative. Citation: Analysis and Applications PubDate: 2021-11-09T08:00:00Z DOI: 10.1142/S0219530521500184
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Authors:Dan Dai, Luming Yao Pages: 1 - 43 Abstract: Analysis and Applications, Ahead of Print. In this paper, we consider the discrete Laguerre polynomials [math] orthogonal with respect to the weight function [math] supported on the infinite nodes [math]. We focus on the “band-saturated region” situation when the parameter [math]. As [math], uniform expansions for [math] are achieved for [math] in different regions in the complex plane. Typically, the Airy-function expansions and Gamma-function expansions are derived for [math] near the endpoints of the band and the origin, respectively. The asymptotics for the normalizing coefficient [math], recurrence coefficients [math] and [math], are also obtained. Our method is based on the Deift–Zhou steepest descent method for Riemann–Hilbert problems. Citation: Analysis and Applications PubDate: 2021-10-20T07:00:00Z DOI: 10.1142/S0219530521500202
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Authors:Nadiia Derevianko, Gerlind Plonka Pages: 1 - 35 Abstract: Analysis and Applications, Ahead of Print. In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form [math], where the frequency parameters [math] are pairwise distinct. In order to reconstruct [math] we employ a finite set of classical Fourier coefficients of [math] with regard to a finite interval [math] with [math]. For our method, [math] Fourier coefficients [math] are sufficient to recover all parameters of [math], where [math] denotes the order of [math]. The recovery is based on the observation that for [math] the terms of [math] possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [Y. Nakatsukasa, O. Sète and L. N. Trefethen, The AAA Algorithm for rational approximation, SIAM J. Sci. Comput. 40(3) (2018) A1494A1522]. If a sufficiently large set of [math] Fourier coefficients of [math] is available (i.e. [math]), then our recovery method automatically detects the number [math] of terms of [math], the multiplicities [math] for [math], as well as all parameters [math], [math], and [math], [math], [math], determining [math]. Therefore, our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony’s method. Citation: Analysis and Applications PubDate: 2021-10-15T07:00:00Z DOI: 10.1142/S0219530521500196
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Authors:Xiuwei Yin, Guangjun Shen, Jiang-Lun Wu Pages: 1 - 13 Abstract: Analysis and Applications, Ahead of Print. In this paper, we study the stability of quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone. The exponential mean square stability and pathwise exponential stability of the solutions are established. Moreover, under certain hypothesis on the stochastic perturbations, pathwise exponential stability can be derived, without utilizing the mean square stability. Citation: Analysis and Applications PubDate: 2021-08-23T07:00:00Z DOI: 10.1142/S0219530521500172
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Authors:Yuan Shen, Yannian Zuo, Liming Sun, Xiayang Zhang Pages: 1 - 28 Abstract: Analysis and Applications, Ahead of Print. We consider the linearly constrained separable convex optimization problem whose objective function is separable with respect to [math] blocks of variables. A bunch of methods have been proposed and extensively studied in the past decade. Specifically, a modified strictly contractive Peaceman–Rachford splitting method (SC-PRCM) [S. H. Jiang and M. Li, A modified strictly contractive Peaceman–Rachford splitting method for multi-block separable convex programming, J. Ind. Manag. Optim. 14(1) (2018) 397-412] has been well studied in the literature for the special case of [math]. Based on the modified SC-PRCM, we present modified proximal symmetric ADMMs (MPSADMMs) to solve the multi-block problem. In MPSADMMs, all subproblems but the first one are attached with a simple proximal term, and the multipliers are updated twice. At the end of each iteration, the output is corrected via a simple correction step. Without stringent assumptions, we establish the global convergence result and the [math] convergence rate in the ergodic sense for the new algorithms. Preliminary numerical results show that our proposed algorithms are effective for solving the linearly constrained quadratic programming and the robust principal component analysis problems. Citation: Analysis and Applications PubDate: 2021-08-16T07:00:00Z DOI: 10.1142/S0219530521500160
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Authors:Akio Ito Pages: 1 - 66 Abstract: Analysis and Applications, Ahead of Print. This paper deals with a nonlinear system (S) composed of three PDEs and one ODE below: (S) ut = ∇⋅ (du(α(v),w)∇β(w; u)) −∇⋅ (u∇λ(v)),vt = dvΔv + awz,wt = −awz,zt = dzΔz − bz + cu. The system (S) was proposed as one of the mathematical models which describe tumor invasion phenomena with chemotaxis effects. The most important and interesting point is that the diffusion coefficient of tumor cells, denoted by [math], is influenced by both nonlocal effect of a chemical attractive substance, denoted by [math], and the local one of extracellular matrix, denoted by [math]. From this point, the first PDE in (S) contains a nonlinear cross diffusion. Actually, this mathematical setting gives an inner product of a suitable real Hilbert space, which governs the dynamics of the density of tumor cells [math], a quasi-variational structure. Hence, the first purpose in this paper is to make it clear what this real Hilbert space is. After this, we show the existence of strong time local solutions to the initial-boundary problems associated with (S) when the space dimension is [math] by applying the general theory of evolution inclusions on real Hilbert spaces with quasi-variational structures. Moreover, for the case [math] we succeed in constructing a strong time global solution. Citation: Analysis and Applications PubDate: 2021-08-04T07:00:00Z DOI: 10.1142/S0219530521500159
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Authors:M. Melgaard, F. D. Y. Zongo Pages: 1 - 18 Abstract: Analysis and Applications, Ahead of Print. We study the nonlinear, nonlocal, time-dependent partial differential equation i∂tφ = (−Δ + m2 − m)φ − 1 x ∗ φ 2 φ on ℝ3, which is known to describe the dynamics of quasi-relativistic boson stars in the mean-field limit. For positive mass parameter [math] we establish existence of infinitely many (corresponding to distinct energies [math]) traveling solitary waves, [math], with speed [math], where [math] corresponds to the speed of light in our choice of units. These traveling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with [math]) because Lorentz covariance fails. Instead, we study a suitable variational problem for which the functions [math] arise as solutions (called boosted excited states) to a Choquard-type equation in [math], where the negative Laplacian is replaced by the pseudo-differential operator [math] and an additional term [math] enters. Moreover, we give a new proof for existence of boosted ground states. The results are based on perturbation methods in critical point theory. Citation: Analysis and Applications PubDate: 2021-07-26T07:00:00Z DOI: 10.1142/S0219530521500147
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Authors:Yunyou Qian, Dansheng Yu Pages: 1 - 23 Abstract: Analysis and Applications, Ahead of Print. In this paper, we introduce some neural network interpolation operators activated by smooth ramp functions. By using the smoothness of the ramp functions, we can give some useful estimates of the derivatives of the neural networks, which combining with some techniques in approximation theory enable us to establish the converse estimates of approximation by neural networks. We establish both the direct and the converse results of approximation by the new neural network operators defined by us, and thus give the essential approximation rate. To improve the approximation rate for functions of smoothness, we further introduce linear combinations of the new operators. The new combinations interpolate the objective function and its derivative. We also estimate the uniform convergence rate and simultaneous approximation rate by the new combinations. Citation: Analysis and Applications PubDate: 2021-07-09T07:00:00Z DOI: 10.1142/S0219530521500123
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Authors:Long Huang, Ferenc Weisz, Dachun Yang, Wen Yuan Pages: 1 - 50 Abstract: Analysis and Applications, Ahead of Print. Let [math], [math] be the mixed-norm Lebesgue space, and [math] an integrable function. In this paper, via establishing the boundedness of the mixed centered Hardy–Littlewood maximal operator [math] from [math] to itself or to the weak mixed-norm Lebesgue space [math] under some sharp assumptions on [math] and [math], the authors show that the [math]-mean of [math] converges to [math] almost everywhere over the diagonal if the Fourier transform [math] of [math] belongs to some mixed-norm homogeneous Herz space [math] with [math] being the conjugate index of [math]. Furthermore, by introducing another mixed-norm homogeneous Herz space and establishing a characterization of this Herz space, the authors then extend the above almost everywhere convergence of [math]-means to the unrestricted case. Finally, the authors show that the [math]-mean of [math] converges over the diagonal to [math] at all its [math]-Lebesgue points if and only if [math] belongs to [math], and a similar conclusion also holds true for the unrestricted convergence at strong [math]-Lebesgue points. Observe that, in all these results, those Herz spaces to which [math] belongs prove to be the best choice in some sense. Citation: Analysis and Applications PubDate: 2021-06-29T07:00:00Z DOI: 10.1142/S0219530521500135
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Authors:Helmut Abels, Christine Pfeuffer Pages: 1 - 36 Abstract: Analysis and Applications, Ahead of Print. In this paper, we show the invariance of the Fredholm index of non-smooth pseudodifferential operators with coefficients in Hölder spaces. By means of this invariance, we improve previous spectral invariance results for non-smooth pseudodifferential operators [math] with coefficients in Hölder spaces. For this purpose, we approximate [math] with smooth pseudodifferential operators and use a spectral invariance result of smooth pseudodifferential operators. Then, we get the spectral invariance result in analogy to a proof of the spectral invariance result for non-smooth differential operators by Rabier. Citation: Analysis and Applications PubDate: 2021-06-26T07:00:00Z DOI: 10.1142/S0219530521500111
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Authors:Mourad E. H. Ismail, Ruiming Zhang Pages: 1 - 18 Abstract: Analysis and Applications, Ahead of Print. In this paper, we use an identity connecting a modified [math]-Bessel function and a [math] function to give [math]-versions of entries in the Lost Notebook of Ramanujan. We also establish an identity which gives an [math]-version of a partition identity. We prove new relations and identities involving theta functions, the Ramanujan function, the Stieltjes–Wigert, [math]-Lommel and [math]-Bessel polynomials. We introduce and study [math]-analogues of the spherical Bessel functions. Citation: Analysis and Applications PubDate: 2021-05-31T07:00:00Z DOI: 10.1142/S0219530521500093
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Authors:Van Tiep Do, Ron Levie, Gitta Kutyniok Pages: 1 - 50 Abstract: Analysis and Applications, Ahead of Print. Natural images are often the superposition of various parts of different geometric characteristics. For instance, an image might be a mixture of cartoon and texture structures. In addition, images are often given with missing data. In this paper, we develop a method for simultaneously decomposing an image to its two underlying parts and inpainting the missing data. Our separation–inpainting method is based on an [math] minimization approach, using two dictionaries, each sparsifying one of the image parts but not the other. We introduce a comprehensive convergence analysis of our method, in a general setting, utilizing the concepts of joint concentration, clustered sparsity, and cluster coherence. As the main application of our theory, we consider the problem of separating and inpainting an image to a cartoon and texture parts. Citation: Analysis and Applications PubDate: 2021-05-07T07:00:00Z DOI: 10.1142/S021953052150007X
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Authors:Xin Zhong Pages: 1 - 27 Abstract: Analysis and Applications, Ahead of Print. We investigate an initial boundary value problem of two-dimensional nonhomogeneous heat conducting magnetohydrodynamic equations. We prove that there exists a unique global strong solution. Moreover, we also obtain the large time decay rates of the solution. Note that the initial data can be arbitrarily large and the initial density allows vacuum states. Our method relies upon the delicate energy estimates and Desjardins’ interpolation inequality (B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal. 137(2) (1997) 135–158). Citation: Analysis and Applications PubDate: 2021-04-30T07:00:00Z DOI: 10.1142/S0219530521500056
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Authors:Shouyou Huang, Yunlong Feng, Qiang Wu Pages: 1 - 19 Abstract: Analysis and Applications, Ahead of Print. Minimum error entropy (MEE) is an information theoretic learning approach that minimizes the information contained in the prediction error, which is measured by entropy. It has been successfully used in various machine learning tasks for its robustness to heavy-tailed distributions and outliers. In this paper, we consider its use in nonparametric regression and analyze its generalization performance from a learning theory perspective by imposing a [math]th order moment condition on the noise variable. To this end, we establish a comparison theorem to characterize the relation between the excess generalization error and the prediction error. A relaxed Bernstein condition and concentration inequalities are used to derive error bounds and learning rates. Note that the [math]th moment condition is rather weak particularly when [math] because the noise variable does not even admit a finite variance in this case. Therefore, our analysis explains the robustness of MEE in the presence of heavy-tailed distributions. Citation: Analysis and Applications PubDate: 2021-03-06T08:00:00Z DOI: 10.1142/S0219530521500044
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Authors:Ernesto de Vito, Massimo Fornasier, Valeriya Naumova Pages: 1 - 48 Abstract: Analysis and Applications, Ahead of Print. Despite a variety of available techniques, such as discrepancy principle, generalized cross validation, and balancing principle, the issue of the proper regularization parameter choice for inverse problems still remains one of the relevant challenges in the field. The main difficulty lies in constructing an efficient rule, allowing to compute the parameter from given noisy data without relying either on any a priori knowledge of the solution, noise level or on the manual input. In this paper, we propose a novel method based on a statistical learning theory framework to approximate the high-dimensional function, which maps noisy data to the optimal Tikhonov regularization parameter. After an offline phase where we observe samples of the noisy data-to-optimal parameter mapping, an estimate of the optimal regularization parameter is computed directly from noisy data. Our assumptions are that ground truth solutions of the inverse problem are statistically distributed in a concentrated manner on (lower-dimensional) linear subspaces and the noise is sub-gaussian. We show that for our method to be efficient, the number of previously observed samples of the noisy data-to-optimal parameter mapping needs to scale at most linearly with the dimension of the solution subspace. We provide explicit error bounds on the approximation accuracy from noisy data of unobserved optimal regularization parameters and ground truth solutions. Even though the results are more of theoretical nature, we present a recipe for the practical implementation of the approach. We conclude with presenting numerical experiments verifying our theoretical results and illustrating the superiority of our method with respect to several state-of-the-art approaches in terms of accuracy or computational time for solving inverse problems of various types. Citation: Analysis and Applications PubDate: 2021-01-30T08:00:00Z DOI: 10.1142/S0219530520500220
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