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From projection pursuit to other unsupervised chemometric techniques

JOURNAL OF CHEMOMETRICS 2007年第7-9期21卷 270-279页

作者： Daszykowski, Michal Silesian Univ Dept Chemometr Inst Chem PL-40006 Katowice Poland

The main goal of exploratory data analysis is to reveal clusters of objects, local changes of data density, outlying objects and/or influential sources of data variance. These different aspects of data exploration can sometimes be accomplished simultaneously with the use of one algorithm, the projection algorithm (PA). In this paper, the PA is described and discussed in detail. It is shown that this algorithm can be considered as a general platform to perform principal components analysis (PCA), robust PCA and independent component analysis (ICA). This goal is achieved by optimizing various different projection indices in the PA. Among these indices one can find entropy, variance and robust scale estimator. The present paper can be regarded as tutorial, aiming to provide a better understanding of the projection pursuit approaches (PPs). Copyright (c) 2007 John Wiley & Sons, Ltd.

关键词： latent variables projection robust PICA projection index projection algorithm

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Nonconvex Differential Variational Inequality and State-Dependent Sweeping Process

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2013年第2期159卷 386-398页

作者： Haddad, Tahar Univ Jijel Fac Sci & Informat Lab Math Pures & Appl Jijel 18000 Algeria

We prove the existence of solutions of a differential variational inequality involving a prox-regular set in an infinite dimensional Hilbert space via a new existence result of a non-convex state-dependent sweeping pr... 详细信息

关键词： Differential variational inequality projection algorithm Prox-regular sets Normal cones

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Solution of Navier-Stokes equations in primitive variables by diffuse approximation

COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE II FASCICULE...

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COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE II FASCICULE B-MECANIQUE PHYSIQUE ASTRONOMIE 1998年第2期326卷 117-119页

作者： Couturier, S Sadat, H Ecole Natl Super Mecan & Aerotech UMR 6068 Lab Etud Therm F-86960 Chasseneuil Futuroscope France

A diffuse approximation method for solving Navier-Stokes equations in primitive variables is proposed. The results of a numerical example show the accuracy and efficiency of this approach. (C) Academie des Sciences/El... 详细信息

关键词： diffuse approximation driven cavity projection algorithm

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Global convergence of a non-convex Douglas-Rachford iteration

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JOURNAL OF GLOBAL OPTIMIZATION 2013年第3期57卷 753-769页

作者： Artacho, Francisco J. Aragon Borwein, Jonathan M. Univ Newcastle Ctr Comp Assisted Res Math & Its Applicat CARMA Callaghan NSW 2308 Australia King Abdulaziz Univ Jeddah 21413 Saudi Arabia

We establish a region of convergence for the proto-typical non-convex Douglas-Rachford iteration which finds a point on the intersection of a line and a circle. Previous work on the non-convex iteration Borwein and Sims (Fixed-point algorithms for inverse problems in science and engineering, pp. 93-109, 2011) was only able to establish local convergence, and was ineffective in that no explicit region of convergence could be given.

关键词： Non-convex feasibility problem Fixed point theory projection algorithm Douglas-Rachford algorithm Global convergence Signal reconstruction

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projection methods for high numerical aperture phase retrieval

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INVERSE PROBLEMS 2021年第12期37卷 125005-125005页

作者： Nguyen Hieu Thao Soloviev, Oleg Luke, Russell Verhaegen, Michel Delft Univ Technol Delft Ctr Syst & Control NL-2628 CD Delft Netherlands Can Tho Univ Sch Educ Dept Math Can Tho Vietnam Flexible Opt BV Polakweg 10-11 NL-2288 GG Rijswijk Netherlands Univ Gottingen Inst Numer & Angew Math D-37083 Gottingen Germany

We develop for the first time a mathematical framework in which the class of projection algorithms can be applied to high numerical aperture (NA) phase retrieval. Within this framework, we first analyze the basic steps of solving the high-NA phase retrieval problem by projection algorithms and establish the closed forms of all the relevant projection operators. We then study the geometry of the high-NA phase retrieval problem and the obtained results are subsequently used to establish convergence criteria of projection algorithms in the presence of noise. Making use of the vectorial point-spread-function (PSF) is, on the one hand, the key difference between this paper and the literature of phase retrieval mathematics which deals with the scalar PSF. The results of this paper, on the other hand, can be viewed as extensions of those concerning projection methods for low-NA phase retrieval. Importantly, the improved performance of projection methods over the other classes of phase retrieval algorithms in the low-NA setting now also becomes applicable to the high-NA case. This is demonstrated by the accompanying numerical results which show that available solution approaches for high-NA phase retrieval are outperformed by projection methods.

关键词： phase retrieval high numerical aperture projection algorithm nonconvex optimization inconsistent feasibility

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A new type of energy-preserving integrators for quasi-bi-Hamiltonian systems

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JOURNAL OF MATHEMATICAL CHEMISTRY 2024年第7期62卷 1667-1681页

作者： Liu, Kai Fu, Ting Shi, Wei Zhou, Xuhuan Nanjing Audit Univ Dept Math Nanjing 211815 Jiangsu Peoples R China Yunnan Normal Univ Yunnan Key Lab Modern Analyt Math & Applicat Kunming 650500 Yunnan Peoples R China Natl Univ Def Technol Dept Math Changsha 410003 Hunan Peoples R China Nanjing Tech Univ Coll Math Sci Nanjing 211816 Jiangsu Peoples R China Nanjing Police Univ Dept Basic Courses Nanjing 210023 Jiangsu Peoples R China

Energy-preserving algorithms, as one of the core research areas in numerical ordinary differential equations, have achieved great success by many methods such as symplectic methods and discrete gradient methods. This paper considers the numerical integration of quasi-bi-Hamiltonian systems, which, as a generalization of bi-Hamiltonian systems, can be expressed in two distinct ways: y=P1(y)del H2(y)=1 rho (y)P2(y)del H1(y). The quasi-bi-Hamiltonian systems have two Hamiltonians H1(y) and H2(y). Conventional discrete gradient methods can only preserve one Hamiltonian at a time. In this paper, based on discrete gradient and projection, new energy-preserving integrators that can preserve the two Hamiltonians simultaneously are proposed. They show better qualitative behaviours than traditional discrete gradient methods do. Numerical integrations of Henon-Heiles type systems and the Korteweg-de Vries (KdV) equation are conducted to show the effectiveness of the new integrators in comparison with traditional discrete gradient methods.

关键词： Quasi-bi-Hamiltonian systems Energy-preserving integrators The discrete gradient methods projection algorithm Henon-Heiles type systems Korteweg-de Vries equation 65L05 65L07 65L20 65P10

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Phase Retrieval with Sparse Phase Constraint

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SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE 2020年第1期2卷 246-263页

作者： Nguyen Hieu Thao Luke, David Russell Soloviev, Oleg Verhaegen, Michel Delft Univ Technol Delft Ctr Syst & Control NL-2628 CD Delft Netherlands Can Tho Univ Sch Educ Dept Math Can Tho Vietnam Univ Gottingen Inst Numer & Angew Math D-37083 Gottingen Germany Flexible Opt BV NL-2288 GG Rijswijk Netherlands

In this paper, we propose and investigate the phase retrieval problem with the a priori constraint that the phase is sparse (SPR), which encompasses a number of practical applications, for instance, in characterizing phase-only objects such as microlenses, in phase-contrast microscopy, in optical path difference microscopy, and in Fourier ptychography, where the phase object occupies a small portion of the whole field. The considered problem is strictly more general than the sparse signal recovery problem, which assumes the sparsity of the signal because the sparsity of the signal trivially implies the sparsity of the phase, but the converse is not true. As a result, existing solution algorithms in the literature of sparse signal recovery cannot be applied to SPR and there is an appeal for developing new solution methods for it. In this paper, we propose a new regularization scheme which efficiently captures the sparsity constraint of SPR. The idea behind the proposed approach is to perform a metric projection of the current estimated signal onto the set of all the signals whose phase satisfies the sparsity constraint. The main challenge here is that the latter set is not convex and its associated projector in general does not admit a closed form. One novelty of our analysis is to establish an explicit form of that projector when restricted to those points which are relevant to the solutions of SPR. Note that this result is fundamentally different from the widely known calculation form for projections onto intensity constraint sets. Based on this new result, we propose an efficient solution method, named the sparsity regularization on phase (SROP) algorithm, for the SPR problem in the challenging setting where only one point-spread-function image is given, and we analyze its convergence. The algorithm is the combination of the Gerchberg-Saxton (GS) algorithm with the projection step described above. In view of the GS algorithm being equivalent to the alternating pro

关键词： phase retrieval sparsity constraint nonconvex optimization prox-regularity projection algorithm linear convergence

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Combining recursive projection and dynamic programming technique in multi UAVs formation anomaly detection

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INTERNATIONAL JOURNAL OF MODELLING IDENTIFICATION AND CONTROL 2019年第1期31卷 53-61页

作者： Wang, Jian-Hong Jiang Univ Sci & Technol Sch Elect & Automat Ganzhou 341000 Peoples R China

To deal with the problem of anomaly detection in multi UAVs formation, and simplify the complexity of hypothesis testing or probability inequalities, the anomaly detection problem can be transformed to identify some unknown parameters process. To avoid a statistical description of measurement noise, a worthwhile alternative is the bounded noise characterisation. In the presence of bounded noise, the projection algorithm with dead zone and its modified form are proposed to identify the unknown parameters, such that the robustness of projection algorithm can be enhanced by increasing a dead zone. Furthermore, dynamic programming technique is introduced to balance the desire for lower present cost with the undesirability of high future cost in determining the anomaly detector, then the cost of collecting new observations and the higher probability of accepting the wrong hypothesis can be compensated. A numerical example illustrates the characteristics of the anomaly detection problem.

关键词： multi UAVs formation anomaly detection projection algorithm dynamic programming dead zone

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On the convergence of general projection methods for solving convex feasibility problems with applications to the inverse problem of image recovery

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OPTIMIZATION 2018年第9期67卷 1409-1427页

作者： Zhao, Xiaopeng Koebis, Markus Arthur Tianjin Polytech Univ Dept Math Tianjin Peoples R China Free Univ Berlin Dept Math & Comp Sci Berlin Germany

For an arbitrary family of closed convex sets with nonempty intersection in a Hilbert space, we consider the classical convex feasibility problem. We study the convergence property of the recently introduced unified projection algorithm B-EMOPP for solving this problem. For this, a new general control strategy is proposed, which we call the 'quasi-coercive control'. Under mild assumptions, we prove the convergence of B-EMOPP using these new control strategies as well as various other strategies. Several known results are extended and improved. The proposed algorithm is then applied to the inverse problem of image recovery.

关键词： Convex feasibility problem image recovery projection algorithm system of closed convex sets

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Model⁃Free Predictive Control for a Kind of High Order Nonlinear Systems

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Journal of Harbin Institute of Technology(New Series) 2022年第2期29卷 62-69页

作者： Ye Tian Baili Su School of Engineering College Qufu Normal UniversityRizhao 276826ShandongChina

For a type of high⁃order discrete⁃time nonlinear systems(HDNS)whose system models are undefined,a model⁃free predictive control(MFPC)algorithm is proposed in this *** first,an estimation model is given by the improved projection algorithm to approach the controlled nonlinear ***,on the basis of the estimation model,a predictive controller is designed by solving the finite time domain rolling optimization quadratic function,and the controller’s explicit analytic solution is also ***,the closed⁃loop system's stability can be ***,the results of simulation reveal that the presented control strategy has a faster convergence speed as well as more stable dynamic property compared with the model⁃free sliding mode control(MFSC).

关键词： nonlinear system compact dynamic linearization(CDL) model predictive control(MPC) model-free control(MFC) projection algorithm

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