An Interpretable Denoising Layer for Neural Networks Based on Reproducing Kernel Hilbert Space and its Application in Machine Fault Diagnosis

Deep learning algorithms based on neural networks make remarkable achievements in machine fault diagnosis,while the noise mixed in measured signals harms the prediction accuracy of networks.Existing denoising methods in neural networks,such as using complex network architectures and introducing sparse techniques,always suffer from the difficulty of estimating hyperparameters and the lack of physical interpretability.To address this issue,this paper proposes a novel interpretable denoising layer based on reproducing kernel Hilbert space(RKHS)as the first layer for standard neural networks,with the aim to combine the advantages of both traditional signal processing technology with physical interpretation and network modeling strategy with parameter adaption.By investigating the influencing mechanism of parameters on the regularization procedure in RKHS,the key parameter that dynamically controls the signal smoothness with low computational cost is selected as the only trainable parameter of the proposed layer.Besides,the forward and backward propagation algorithms of the designed layer are formulated to ensure that the selected parameter can be automatically updated together with other parameters in the neural network.Moreover,exponential and piecewise functions are introduced in the weight updating process to keep the trainable weight within a reasonable range and avoid the ill-conditioned problem.Experiment studies verify the effectiveness and compatibility of the proposed layer design method in intelligent fault diagnosis of machinery in noisy environments.

RECURSIVE REPRODUCING KERNELS HILBERT SPACES USING THE THEORY OF POWER KERNELS

The main objective of this work is to decompose orthogonally the reproducing kernels Hilbert space using any conditionally positive definite kernels into smaller ones by introducing the theory of power kernels, and to show how to do this decomposition recur- sively. It may be used to split large interpolation problems into smaller ones with different kernels which are related to the original kernels. To reach this objective, we will reconstruct the reproducing kernels Hilbert space for the normalized and the extended kernels and give the recursive algorithm of this decomposition.

Minimax designs for linear regression models with bias in a reproducing kernel Hilbert space in a discrete set

Consider the design problem for estimation and extrapolation in approximately linear regression models with possible misspecification. The design space is a discrete set consisting of finitely many points, and the model bias comes from a reproducing kernel Hilbert space. Two different design criteria are proposed by applying the minimax approach for estimating the parameters of the regression response and extrapolating the regression response to points outside of the design space. A simulated annealing algorithm is applied to construct the minimax designs. These minimax designs are compared with the classical D-optimal designs and all-bias extrapolation designs. Numerical results indicate that the simulated annealing algorithm is feasible and the minimax designs are robust against bias caused by model misspecification.

ON APPROXIMATION BY SPHERICAL REPRODUCING KERNEL HILBERT SPACES

The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the subspace with better smoothness. Furthermore, the upper bound of approximation error is given.

The Expansion of the Function with Two Unknowns on the Reproducing Kernel Space

In this paper we make use of a special procedure on the repro ducing kernel space to give an expansion theorem for the function with two unkno wns and a surface approximation formula. The error of the surface possesses mono tonically decreasing and uniformly convergent characteristics in the sense of t he norm on the space.

New Implementation of Reproducing Kernel Method for Solving Functional-Differential Equations

In this paper, we apply the new algorithm of reproducing kernel method to give the approximate solution to some functional-differential equations. The numerical results demonstrate the accuracy of the proposed algorithm.

A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

In this paper,the weak pre-orthogonal adaptive Fourier decomposition(W-POAFD)method is applied to solve fractional boundary value problems(FBVPs)in the reproducing kernel Hilbert spaces(RKHSs)W_(0)^(4)[0,1] and W^(1)[0,1].The process of the W-POAFD is as follows:(i)choose a dictionary and implement the pre-orthogonalization to all the dictionary elements;(ii)select points in[0,1]by the weak maximal selection principle to determine the corresponding orthonormalized dictionary elements iteratively;(iii)express the analytical solution as a linear combination of these determined dictionary elements.Convergence properties of numerical solutions are also discussed.The numerical experiments are carried out to illustrate the accuracy and efficiency of W-POAFD for solving FBVPs.

Attractive Multistep Reproducing Kernel Approach for Solving Stiffness Differential Systems of Ordinary Differential Equations and Some Error Analysis

In this paper,an efficient multi-step scheme is presented based on reproducing kernel Hilbert space(RKHS)theory for solving ordinary stiff differential systems.The solution methodology depends on reproducing kernel functions to obtain analytic solutions in a uniform formfor a rapidly convergent series in the posed Sobolev space.Using the Gram-Schmidt orthogonality process,complete orthogonal essential functions are obtained in a compact field to encompass Fourier series expansion with the help of kernel properties reproduction.Consequently,by applying the standard RKHS method to each subinterval,approximate solutions that converge uniformly to the exact solutions are obtained.For this purpose,several numerical examples are tested to show proposed algorithm’s superiority,simplicity,and efficiency.The gained results indicate that themulti-step RKHSmethod is suitable for solving linear and nonlinear stiffness systems over an extensive duration and giving highly accurate outcomes.

Convergence analysis for complementary-label learning with kernel ridge regression

Complementary-label learning(CLL)aims at finding a classifier via samples with complementary labels.Such data is considered to contain less information than ordinary-label samples.The transition matrix between the true label and the complementary label,and some loss functions have been developed to handle this problem.In this paper,we show that CLL can be transformed into ordinary classification under some mild conditions,which indicates that the complementary labels can supply enough information in most cases.As an example,an extensive misclassification error analysis was performed for the Kernel Ridge Regression(KRR)method applied to multiple complementary-label learning(MCLL),which demonstrates its superior performance compared to existing approaches.

Reproducing wavelet kernel method in nonlinear system identification

By combining the wavelet decomposition with kernel method, a practical approach of universal multiscale wavelet kernels constructed in reproducing kernel Hilbert space （RKHS） is discussed, and an identification scheme using wavelet support vector machines （WSVM） estimator is proposed for nordinear dynamic systems. The good approximating properties of wavelet kernel function enhance the generalization ability of the proposed method, and the comparison of some numerical experimental results between the novel approach and some existing methods is encouraging.

Solving Neumann Boundary Problem with Kernel-Regularized Learning Approach

We provide a kernel-regularized method to give theory solutions for Neumann boundary value problem on the unit ball. We define the reproducing kernel Hilbert space with the spherical harmonics associated with an inner product defined on both the unit ball and the unit sphere, construct the kernel-regularized learning algorithm from the view of semi-supervised learning and bound the upper bounds for the learning rates. The theory analysis shows that the learning algorithm has better uniform convergence according to the number of samples. The research can be regarded as an application of kernel-regularized semi-supervised learning.

A Gradient Iteration Method for Functional Linear Regression in Reproducing Kernel Hilbert Spaces

We consider a gradient iteration algorithm for prediction of functional linear regression under the framework of reproducing kernel Hilbert spaces.In the algorithm,we use an early stopping technique,instead of the classical Tikhonov regularization,to prevent the iteration from an overfitting function.Under mild conditions,we obtain upper bounds,essentially matching the known minimax lower bounds,for excess prediction risk.An almost sure convergence is also established for the proposed algorithm.

MEAN VALUE OF ANALYTIC OPERATOR FUNCTIONS

In this paper, it is proved that if A is a normal proper contraction on Hilbert space H and F(z) = U(z) + iV(z) is operator-valued analytic on the unit disc Delta and 0 < p < 1, then parallel to F(A)parallel to(p) less than or equal to parallel to F(0)parallel to(p) + C-p (1 - parallel to A parallel to)(-2) [GRAPHICS]

基于核机器的加速失效时间模型及其应用

加速失效时间模型是一种应用广泛的生存分析模型。本文借助LASSO惩罚剔除冗余预测变量,构建基于核机器的加速失效时间模型,用以刻画预测变量与生存期间的复杂关系。此外,提出一种新的正则化Garrotized核机器估计方法,可以较好地刻画预测变量与生存期潜在的非线性关系,实现非参数分量中预测变量间交互作用的自动建模,提升模型预测精度。模拟研究表明,与已有的代表性方法相比,本文提出的方法对生存期的预测精度更高,特别是在复杂关系情形下优势更为显著。最后,将该方法应用于胃癌数据分析,利用临床信息和基因表达预测生存期和风险评分。实证结果显示,该方法能为病例基于风险分层的临床精准诊疗方案设计提供有益的参考。

Large Dynamic Covariance Matrix Estimation with an Application to Portfolio Allocation:A Semiparametric Reproducing Kernel Hilbert Space Approach

The estimation of high dimensional covariance matrices is an interesting and important research topic for many empirical time series problems such as asset allocation. To solve this dimension dilemma, a factor structure has often been taken into account. This paper proposes a dynamic factor structure whose factor loadings are generated in reproducing kernel Hilbert space(RKHS), to capture the dynamic feature of the covariance matrix. A simulation study is carried out to demonstrate its performance. Four different conditional variance models are considered for checking the robustness of our method and solving the conditional heteroscedasticity in the empirical study. By exploring the performance among eight introduced model candidates and the market baseline, the empirical study from 2001 to 2017 shows that portfolio allocation based on this dynamic factor structure can significantly reduce the variance, i.e., the risk, of portfolio and thus outperform the market baseline and the ones based on the traditional factor model.

基于再生核Hilbert空间小波核函数支持向量机的高光谱遥感影像分类

针对支持向量机用于高光谱遥感影像分类存在的分类精度不高、参数选择困难等问题,提出一种再生核Hilbert空间的小波核。其可以逼近任意非线性函数,能够有效改进参数估计的效果,进而实现基于再生核Hilbert空间的小波核函数支持向量机(小波支持向量机)。并选取北京昌平地区的国产高光谱数据operational modular imaging spec-trometer II(OMIS II)和意大利Pavia大学ROSIS高光谱数据进行试验。结果表明,应用Coiflet小波核函数时能获得较高分类精度。

一个具有混合核的Hilbert型积分不等式及推广

利用权函数的方法,建立一个新的、具有混合核的Hilbert型积分不等式,并证明其常数因子为最佳值,还考虑了其等价式及(p,q)-参数形式的最佳推广.

Differentially private SGD with random features

In the realm of large-scale machine learning,it is crucial to explore methods for reducing computational complexity and memory demands while maintaining generalization performance.Additionally,since the collected data may contain some sensitive information,it is also of great significance to study privacy-preserving machine learning algorithms.This paper focuses on the performance of the differentially private stochastic gradient descent(SGD)algorithm based on random features.To begin,the algorithm maps the original data into a lowdimensional space,thereby avoiding the traditional kernel method for large-scale data storage requirement.Subsequently,the algorithm iteratively optimizes parameters using the stochastic gradient descent approach.Lastly,the output perturbation mechanism is employed to introduce random noise,ensuring algorithmic privacy.We prove that the proposed algorithm satisfies the differential privacy while achieving fast convergence rates under some mild conditions.

THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS

This study introduces a pre-orthogonal adaptive Fourier decomposition(POAFD)to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre(generalized Poisson equation).As a first step,the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence,and,as a second step,makes use of the semigroup or the reproducing kernel property of each of the expanding entries.Experiments show the effectiveness and efficiency of the proposed series solutions.

基于再生核Hilbert空间的非线性信道均衡算法

在高速无线通信领域,为消除码间干扰(ISI)必须研究非线性信道均衡技术。基于再生核希尔伯特空间(RKHS)研究非线性信道的自适应均衡算法。首先基于非线性维纳模型提出均衡器的结构,基于RKHS引入核方法,与仿射投影算法(APA)相结合推导出核仿射投影算法(KAPA),再通过引入松弛因子得到改进的KAPA算法。用蒙特卡罗法对提出的自适应算法进行仿真,从收敛性能、误码率(BER)、跟踪能力、计算复杂度等方面与其他算法做比较。在不增加计算复杂度的情况下,极大降低了误码率,非常适合时变非线性信道均衡的应用。