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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.
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17 citations
TL;DR: In this article, a linear-time algorithm for approximating a set of n points by a linear function, or a line, that minimizes the L 1 norm is presented, which is optimal within a constant factor and enables to use linearL 1 approximation of many points in practice.
Abstract: In this paper we present a linear-time algorithm for approximating a set ofn points by a linear function, or a line, that minimizes theL1 norm. The algorithmic complexity of this problem appears not to have been investigated, although anO(n3) naive algorithm can be easily obtained based on some simple characteristics of an optimumL1 solution. Our linear-time algorithm is optimal within a constant factor and enables us to use linearL1 approximation of many points in practice. The complexity ofL1 linear approximation of a piecewise linear function is also touched upon.
17 citations
TL;DR: A piecewise linear approximation to the unknown functions during the system operation is developed to achieve semiglobal stability of the state, accurate tracking of bounded reference signals contained within a known domain, and at least boundedness of the function approximator parameter estimates.
Abstract: This paper is concerned with tracking control problems for nonlinear systems that are not affine in the control signal and that contain unknown nonlinearities in the system dynamic equations. This paper develops a piecewise linear approximation to the unknown functions during the system operation. New control and parameter adaptation algorithms are designed and analyzed using Lyapunov-like methods. The objectives are to achieve semiglobal stability of the state, accurate tracking of bounded reference signals contained within a known domain , and at least boundedness of the function approximator parameter estimates. Numerical simulations are included to illustrate the effectiveness of the learning algorithm.
17 citations
10 Dec 2006
TL;DR: In this paper, the rank-2 property is considered in the initialization of a non-linear 8-point method, and the proposed method minimizes the algebraic error, making sure that the results have the rank 2 property, which is neglected in the conventional linear method.
Abstract: Given a set of point correspondences in an uncalibrated image pair, we can estimate the fundamental matrix, which can be used in calculating several geometric properties of the images. Among the several existing estimation methods, nonlinear methods can yield accurate results if an approximation to the true solution is given, whereas linear methods are inaccurate but no prior knowledge about the solution is required. Usually a linear method is employed to initialize a nonlinear method, but this sometimes results in failure when the linear approximation is far from the true solution. We herein describe an alternative, or complementary, method for the initialization. The proposed method minimizes the algebraic error, making sure that the results have the rank-2 property, which is neglected in the conventional linear method. Although an approximation is still required in order to obtain a feasible algorithm, the method still outperforms the conventional linear 8-point method, and is even comparable to Sampson error minimization.
16 citations
TL;DR: A convexity analysis on the anisotropic interfacial energy is performed, and a careful estimate reveals that all its second order functional derivatives stay uniformly bounded by a global constant, which enables one to derive an energy stable numerical scheme.
16 citations