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Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.


Papers
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Journal ArticleDOI
TL;DR: Though this framework is completely algorithmic, it provides solutions with optimal statistical performances and controlled algorithmic complexity for a large family of nonconvex optimization problems.
Abstract: We propose a computational framework named iterative local adaptive majorize-minimization (I-LAMM) to simultaneously control algorithmic complexity and statistical error when fitting high dimensional models. I-LAMM is a two-stage algorithmic implementation of the local linear approximation to a family of folded concave penalized quasi-likelihood. The first stage solves a convex program with a crude precision tolerance to obtain a coarse initial estimator, which is further refined in the second stage by iteratively solving a sequence of convex programs with smaller precision tolerances. Theoretically, we establish a phase transition: the first stage has a sublinear iteration complexity, while the second stage achieves an improved linear rate of convergence. Though this framework is completely algorithmic, it provides solutions with optimal statistical performances and controlled algorithmic complexity for a large family of nonconvex optimization problems. The iteration effects on statistical errors are clearly demonstrated via a contraction property. Our theory relies on a localized version of the sparse/restricted eigenvalue condition, which allows us to analyze a large family of loss and penalty functions and provide optimality guarantees under very weak assumptions (For example, I-LAMM requires much weaker minimal signal strength than other procedures). Thorough numerical results are provided to support the obtained theory.

85 citations

Journal ArticleDOI
TL;DR: In this article, necessary and sufficient conditions for affine nonlinear systems to be globally feedback equivalent to a controllable linear system over an open subset V of Rn are presented, when V equals Rn.
Abstract: This note presents necessary conditions and sufficient conditions for an affine nonlinear system to be globally feedback equivalent to a controllable linear system over an open subset V of Rn. When V equals Rn, necessary and sufficient conditions are obtained.

85 citations

Book
17 Sep 2004
TL;DR: The best ebooks about Approximation Theory Using Positive Linear Operators that you can get for free here by download this approximation theory using positive linear operators and save to your desktop.
Abstract: The best ebooks about Approximation Theory Using Positive Linear Operators that you can get for free here by download this Approximation Theory Using Positive Linear Operators and save to your desktop. This ebooks is under topic such as approximation theory using positive linear operators statistical fuzzy approximation theory by fuzzy positive approximation theory using positive linear operators a-summation process and korovkin-type approximation approximation theory using positive linear operators uniform weighted approximation by positive linear operators approximation theory using positive linear operators approximation by certain positive linear operators utcluj statistical approximation by positive linear operators on the a-statistical approximation by sequences of k uniform approximation in weighted spaces using some approximation of analytical functions by sequences of k statistical approximation properties of a generalization 1 maximum likelihood estimation of functionals of discrete approximation of functions of two variables by some linear approximation of functions of two variables by some linear weighted approximation by positive linear operators contributions to the approximation of functions evaluation of the approximation order by positive linear statistical convergence applied to korovkin-type higher order generalization of positive linear operators on linear and positive operators wseas statistical approximation for new positive linear i−convergence theorems for a class of k-positive linear rates of ideal convergence for approximation operators a note on the statistical approximation properties of the matrix summability and positive linear operators ozlem g local approximation results for sz ́asz-mirakjan type operators a korovkin-type approximation theorem for double sequences approximation theory and functional analysis on time scales some approximation theorems for a general class of prof dr radu p alt anea transilvania university of braÈÂTMov approximation of functions by some types of szasz-mirakjan some approximation results for bernstein-kantorovich approximation by a generalization of the arxiv approximation by kantorovich-szász type operators based on approximation of functions by convexity ams on approximation properties of certain multidimensional approximation properties of rth order generalized

85 citations

Journal ArticleDOI
TL;DR: The main features of the method are the following: rapid convergence on the entire representative set of parameters, rigorous a posteriori error estimators for the output, and a parameter independent off-linephase and a computationally very efficient on-line phase to enable the rapid solution of many-query problems arising in control, optimization, and design.
Abstract: We propose certified reduced basis methods for the efficient and reliable evaluation of a general output that is implicitly connected to a given parameterized input through the harmonic Maxwell's equations. The truth approximation and the development of the reduced basis through a greedy approach is based on a discontinuous Galerkin approximation of the linear partial differential equation. The formulation allows the use of different approximation spaces for solving the primal and the dual truth approximation problems to respect the characteristics of both problem types, leading to an overall reduction in the off-line computational effort. The main features of the method are the following: (i) rapid convergence on the entire representative set of parameters, (ii) rigorous a posteriori error estimators for the output, and (iii) a parameter independent off-line phase and a computationally very efficient on-line phase to enable the rapid solution of many-query problems arising in control, optimization, and design. The versatility and performance of this approach is shown through a numerical experiment, illustrating the modeling of material variations and problems with resonant behavior.

85 citations

Journal ArticleDOI
TL;DR: The proposed S-box is more secure against differential and linear cryptanalysis compared to recently proposed chaotic S-boxes, and has very optimal nonlinearity, bit independence criterion (BIC), strict avalanche criterion (SAC), differential andlinear approximation probabilities.
Abstract: A substitution box (S-box) plays a central role in cryptographic algorithms. In this paper, an efficient method for designing S-boxes based on chaotic maps is proposed. The proposed method is based on the NCA (nonlinear chaotic algorithm) chaotic maps. The S-box so constructed has very optimal nonlinearity, bit independence criterion (BIC), strict avalanche criterion (SAC), differential and linear approximation probabilities. The proposed S-box is more secure against differential and linear cryptanalysis compared to recently proposed chaotic S-boxes.

85 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20237
202229
202197
2020134
2019124
2018147