基于函数型数据的系数正则化回归的收敛速度（英文）

本文研究了基于函数型输入和_1-正则化的最小二乘回归问题的推广性能.利用基于Rademacher平均的分析技术,获得了学习速度的估计,推广了已有的欧式空间有限维输入结果.     

回归函数核估计的收敛速度

本文在P≥1的条件下,给出了回归函数m(x)的核估计m_n(x)的若干种p阶平均收敛速度,改进并推广了文献[1]及[2]中的若干结果。     

函数型数据回归分析综述

随着计算机储存能力和在线观测技术的提高,当今数据越来越多的以曲线和图像的形式存在.曲线和图像数据两个最显著的特征是高维和相邻数据间高度相关.这些特征使得传统的多元统计分析方法不再适合,而函数型数据在处理曲线和图像数据中具有无可比拟的优势.近年来各种各样的函数型数据分析方法得以发展,其中包括数据的对齐、主成分分析、回归、分类、聚类等.本文主要介绍函数型数据回归分析研究的起源、发展及最新进展.具体地,本文首先介绍函数型数据的概念;其次介绍函数型主成分分析方法;再次着重介绍函数型回归模型的估计、变量选择和检验方法;最后将简要探讨函数型数据未来的可能发展方向.     

l~P-系数正则化Shannon采样学习算法收敛速度(英文)

研究l~P-系数正则化意义下Shannon采样学习算法的收敛速度估计问题.借助l~P-空间的凸性不等式给出了样本误差和正则化误差的上界估计,并给出了用K-泛函表示的逼近误差估计.将K-泛函的收敛速度估计转化为平移网络逼近问题,在此基础上给出了用概率表示的学习速度.     

基于相关熵损失的核正则化回归学习速度（英文）

最大相关熵回归在信号处理领域有广泛应用,其收敛性分析是机器学习领域中的热门研究课题.本文给出一种新的误差分析框架,将非凸优化问题转化为局部凸优化问题,然后应用凸分析方法给出最大相关熵回归(MCCR)收敛性的理论分析;将最优化回归函数表示成一种积分方程的解,用K-泛函和再生核Hilbert空间最佳逼近表示泛化误差,给出学习速度的一种上界估计.     

单位球上散乱数据核正则化回归的误差分析

给出基于二次损失的单位球盖(单位球)上确定型散乱数据核正则化回归误差的上界估计,将学习误差估计转化为核函数积分的误差分析,借助于学习理论中的K-泛函与光滑模的等价性刻画了学习速度.研究结果表明学习速度由网格范数所控制.     

非参数回归函数核估计的强收敛速度

本文给出回归函数m(x)=E(Y|X=x)满足λ(0<λ≤1)阶Lipschitz条件,且E|Y|~r<∞,r>1时,对m(x)的核估计有同时本文也改善了赵林城、方兆本(1985年)和孙东初(1985年)关于m_n(x)强相合于m(x)的结果。     

L1正则化Logistic回归在财务预警中的应用

线性模型和广义线性模型已广泛地用于社会经济、生产实践和科学研究中的数据分析和数据挖掘等领域,如公司财务预警,引入L1范数惩罚技术的模型在估计模型系数的同时能实现变量选择的功能,本文将L1范数正则化Logistic回归模型用于上市公司财务危机预报,结合沪深股市制造业ST公司和正常公司的T-2年财务数据开展实证研究,舛比Logistic回归和L2正则化Logistic回归模型进行对比分析.实验结果表明L1正则化Logistic回归模型的有效性,其在保证模型预测精度的同时提高模型的解释性.     

非参数回归函数核估计的Lp收敛速度

在p≥1和适当的条件下，给出了回归函数m(x)＝E（Y|X＝x）的核估计的若干种Lp收敛速度，改进并推广了韦来生（1984）的结果。     

正则化回归算法的快速学习率

本文讨论了再生核Hilbert 空间上一类广泛的正则化回归算法的学习率问题. 在分析算法的样本误差时, 我们利用了一种复加权的经验过程, 保证了方差与惩罚泛函同时被阈值控制, 从而避免了繁琐的迭代过程. 本文得到了比之前文献结果更为快速的学习率.     

An algorithm for fitting circular arcs to data using the <Emphasis Type=Italic>l</Emphasis><Subscript><Emphasis Type=Italic>1</Emphasis></Subscript> norm

There are many applications of fitting circular arcs to data. We have for example, system control, using a computer controlled cutting machine, approximating hulls of boats, drawing and image techniques. Out of these applications comes the least squares norm to be the most commonly used criterion. This paper examines how the l ₁ norm is used which seems to be more appropriate than the use of least squares in the context of wild points in the data. An algorithm and different methods to determine the starting points are developed. However, numerical examples are given to help illustrate these methods.      

Controller design to optimize a composite performance measure

This paper studies a mixed objective problem of minimizing a composite measure of thel ₁, ₂, andl -norms together with thel -norm of the step response of the closed loop. This performance index can be used to generate Pareto-optimal solutions with respect to the individual measures. The problem is analyzed for discrete-time, single-input single-output (SISO), linear time-invariant systems. It is shown via Lagrange duality theory that the problem can be reduced to a convex optimization problem with a priori known dimension. In addition, continuity of the unique optimal solution with respect to changes in the coefficients of the linear combination that defines the performance measure is estabilished.This research was supported by the National Science Foundation under Grants No. ECS-92-04309, ECS-92-16690 and ECS-93-08481.     

Benders' decomposition for the constrainedl 1-problem

A constrainedl ₁-problem, involving linear functions only, is considered, and the application of the Benders decomposition method to the solution of the same is discussed. This approach, in principle, seems to be promising and is also applicable to the unconstrained case. Certain small illustrative examples are also presented.This work was done while the first author was visiting the Numerical Optimisation Centre at the Hatfield Polytechnic under the auspices of the British Council and he wishes to acknowledge the excellent research facilities provided by the Polytechnic during the period of his stay. Thanks are also due to Dr. M. C. Bartholomew-Biggs for his valuable comments at various stages of the preparation of this report.     

Fitting hyperplanes by minimizing orthogonal deviations

Hyperplanes withm + 1 parameters are fitted by minimizing the sum of weighted orthogonal deviations to a set ofN points. There is no inverse regression incompatibility. For unweighted orthogonall ₁-fits essentially the same number of points are on either side of an optimal hyperplane. The criterion function is neither convex, nor concave, nor even differentiable. The main result is that each orthogonall _p -fit interpolates at leastm + 1 points, for 0 <p 1. This enables the combinatorial strategy of systematically trying all possible hyperplanes which interpolatem + 1 data points.     

Two Characterizations of the Standard Unit Vector Basis of l 1

We show that for a sequence in a Banach space, the property of being stable under large perturbations characterizes the property of being equivalent to the unit vector basis of l ₁. We show that a normalized unconditional basic sequence in l ₁ that is semi-normalized in l is equivalent to the standard unit vector basis of l ₁.     

A Variation of an Extremal Theorem Due to Woodall

We consider a variation of an extremal theorem due to Woodall [12, or 1, Chapter 3] as follows: Determine the smallest even integer (₃C₁,n), such that every n-term graphic sequence = (d₁, d₂,..., d_n) with term sum () = d₁ + d₂ + ... + d_n (₃C₁,n) has a realization G containing a cycle of length r for each r = 3,4,...,l. In this paper, the values of (₃C_l,n) are determined for l = 2m – 1,n 3m – 4 and for l = 2m,n 5m – 7, where m 4.AMS Mathematics subject classification (1991) 05C35Project supported by the National Natural Science Foundation of China (Grant No. 19971086) and the Doctoral Program Foundation of National Education Department of China     

On solving three classes of nonlinear programming problems via simple differentiable penalty functions

We consider the following classes of nonlinear programming problems: the minimization of smooth functions subject to general constraints and simple bounds on the variables; the nonlinearl ₁-problem; and the minimax problem. Numerically reliable methods for solving problems in each of these classes, based upon exploiting the structure of the problem in constructing simple differentiable penalty functions, are presented.This research was made possible by NSERC Grant No. A8442.The author would like to thank Mrs. J. Selwood of the Department of Combinatories and Optimization, University of Waterloo, Ontario, Canada for her excellent typesetting.This work was carried out in the Department of Combinatories and Optimization, University of Waterloo, Waterloo, Ontario, Canada.     

Convergence of a Reactive Planning Algorithm

The reactive planning algorithms proposed by P. Maes in Artificial Intelligence for real-time decision making and other derived algorithms are empirically known to converge, but no mathematical proof has been proposed. The evolution equations of these incremental algorithms can be translated into discrete time non-linear dynamical systems. We prove the convergence of such dynamical systems, provided the necessary assumptions are fulfilled, and we give an expression for the limit.     

Multiple criteria linear regression

The unknown parameters in multiple linear regression models may be estimated using any one of a number of criteria such as the minimization of the sum of squared errors MSSE, the minimization of the sum of absolute errors MSAE, and the minimization of the maximum absolute error MMAE. At present, the MSSE or the least squares criterion continues to be the most popular. However, at times the choice of a criterion is not clear from statistical, practical or other considerations. Under such circumstances, it may be more appropriate to use multiple criteria rather than a single criterion to estimate the unknown parameters in a multiple linear regression model. We motivate the use of multiple criteria estimation in linear regression models with an example, propose a few models, and outline a solution procedure.     

Uniformly non-l n (1) Musielak-Orlicz sequence spaces

We give a necessary and sufficient condition for the uniformly non-l _n ⁽¹⁾ property of Musielak-Orlicz sequence spacesl ^Φ generated by a sequence Φ=(ϕ_n:n⩾l) of finite Orlicz functions such that for eachn∈ℕ. As a result, forn ₀⩾2, there exist spacesl ^Φ which are only uniformly non-l _n ⁽¹⁾ forn⩾n ₀. Moreover we obtain a characterization of uniformly non-l _n ⁽¹⁾ and reflexive Orlicz sequence spaces over a wide class of purely atomic measures and of uniformly non-l _n ⁽¹⁾ Nakano sequence spaces. This extends a result of Luxemburg in [19]. Submitted in memory of Professor W. Orlicz