Learning theory: stability is sufficient for generalization and necessary and sufficient for consistency of empirical risk minimization

Solutions of learning problems by Empirical Risk Minimization (ERM) – and almost-ERM when the minimizer does not exist – need to be consistent, so that they may be predictive. They also need to be well-posed in the sense of being stable, so that they might be used robustly. We propose a statistical form of stability, defined as leave-one-out (LOO) stability. We prove that for bounded loss classes LOO stability is (a) sufficient for generalization, that is convergence in probability of the empirical error to the expected error, for any algorithm satisfying it and, (b) necessary and sufficient for consistency of ERM. Thus LOO stability is a weak form of stability that represents a sufficient condition for generalization for symmetric learning algorithms while subsuming the classical conditions for consistency of ERM. In particular, we conclude that a certain form of well-posedness and consistency are equivalent for ERM. Dedicated to Charles A. Micchelli on his 60th birthday Mathematics subject classifications (2000) 68T05, 68T10, 68Q32, 62M20. Tomaso Poggio: Corresponding author.     

Modified pinhole spatial filter producing a clean flat-topped beam

The expanded beam from a laser has a sharply peaked intensity profile. As a result, in many applications where uniform illumination of an extended field is required, it is only possible to use the central part of the beam. We show that if the pinhole normally used to spatially filter the beam is replaced by an annular phase mask, it should be possible to obtain a clean beam providing very nearly uniform illumination over an extended area, with a minimal loss of light.     

The Homomorphism and Isomorphism on Hypergroup

61.IntroductionThispapersupposesthatthe(G,')isagroup,inP,(G)HP(G)-{di},itisstiuplatedtheoperationasthefollowing:A.BH{a.blaeA,beB}.(1.1)ThenP,(G)isasemigroup.Iftheunempty@=P,(G)makesupagroupwith(1.l),then@iscalledthehypergroupofG,here'E'standsfortheunitelement,especially,wheneeEinG,wecallgasareguIarhypergroup,whencontraryelementsandcontrarysetsof@conformedtoeachother,wecallopasauniformhypergroup.Obviously,Eisasemi-subgroupofG.Whengisregular,Eisasemi-subgroupwithunitelement,When@isunif…     

Solutions of system of generalized vector quasi-equilibrium problems in locally G-convex uniform spaces

In this paper we establish a collectively fixed point theorem and an equilibrium existence theorem for generalized games in product locally G-convex uniform spaces. As applications, some new existence theorems of solutions for the system of generalized vector quasi-equilibrium problems are derived in product locally G-convex uniform spaces. These theorems are new and generalize some known results in the literature.     

On the uniform boundedness of solutions of some non-autonomous differential equations of the fourth order

In this paper, sufficient conditions are established under which all solutions of some non-autonomous differential equations of the fourth order are bounded and uniformly bounded.     

On a New Summability Result for Operator Matrices

For matrices of quasi-homogeneous operators we establish a summability result which is a useful complement of a recent summability result.     

照明系统中点光源的光照度分布分析

简要分析了照明光学系统与成像光学系统的区别。介绍了目前光学设计软件的分类。运用一个具体的照明光学系统分析了点光源的光照度分布分析软件的特性。     

Application of the structured illumination method to study the topography of the sole of the foot during a walk

The structured illumination technique consists of projecting a fringe system onto a 3D object from a well defined space point, which results in a pattern that depends on the characteristic of the projected fringes, the viewpoint and the illuminated object morphology. Therefore, the structured illumination method enables to determine the topography of 3D objects. To implement this technique we set up an optoelectronic array that allows studying the sole of the foot during a walk. The method consists of projecting a Ronchi grid and capturing the images generated on the foot by a CCD camera. These obtained images are graphically processed and fringes converted into vectors. A depending algorithm on the experimental setup allows obtaining, from those vectors data, cotes for a discrete profile plotting of the studied object. The method enables the quantitative determination of the sole topography during the walk. Qualitatively, it can be used for diagnosis and control of deformation and injuries caused by accidents or illnesses. It can be introduced in the primary attention health system to study a great number of patients, due to its simplicity and low cost installation, and for being a non-invasive technique.     

SUPERCONVERGENCE OF LEAST-SQUARES MIXED FINITE ELEMENTS FOR ELLIPTIC PROBLEMS ON TRIANGULATION

In this paper, we present the least-squares mixed finite element method and investigate superconvergence phenomena for the second order elliptic boundary-value problems over triangulations. On the basis of the L~2-projection and some mixed finite element projections, we obtain the superconvergence result of least-squares mixed finite     

Multisymplectic Structure-Preserving in Simple Finite Element Method in High Dimensional Case

In this paper, we study a finite element scheme of some semi-linear elliptic boundary value problems inhigh-dimensional space. With uniform mesh, we find that, the numerical scheme derived from finite element method cankeep a preserved multisymplectic structure.