A. Pinkus, n-Widths in Approximation Theory, Springer-Verlag, Berlin, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....model order. Further motivation of this work comes from the analysis of the optimal achievable worst case error by Zames et al. in [ZLW94] and from a subsequent tuned FIR approximation scheme suggested by Glaum, Lin and Zames [GLZ96] The Kolmogorov n width of a subset P#H #,i nH# is defined by [Pin85] d n (P) inf Xn x#P y #Xn #x y## (6.1) where the left most infimum is taken over all n dimensional subspaces of H#.Thusd n (P) represents the optimal worst case error one can achieve by approximating a system in by a linear combination of n basis functions in . A subspace ....

....are interested in two plant sets: 1 (#,#) BH#,# (#)and 2 (M) f : f M (6.2) 45 6.2 Problem Formulation 46 .ForbothP 1 (#, #)andP 2 (M) an optimal subspace is S n = span 1,z, z (6.3) where span is over the real field. Further, the exact values of n widths in these cases are [Pin85] d n (P 1 (#, #) ## n ,d n (P 2 (M) 6.4) Clearly, d n (P i ) 0,i=1, 2. Contrary to intuition however, an optimal approximation is not always given by the first n impulse response parameters. 6.2 Problem Formulation Assumption : The plant transfer function P (z) to be identified ....

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A. Pinkus. n-widths in Approximation Theory. Springer-Verlag, 1985.

....L 2( n Widths From now on we let be a positive Hilbert Schmidt kernel on L 2( and assume (15) to play safe. We make use of the fact that we have integral operators related to (x; y) or (x; y) that map L 2( into N or R . This opens the road for applications of the theory of n widths [11]. For the convenience of the reader, we will review that part that is of interest for us. For a subset A of a Hilbert space H , the Kolmogorov n width is de ned by dn (A; H) inf Vn sup f2H s2Vn kf skH : Here, the outer in mum is taken over all n dimensional subspaces Vn of H . An ....

....ball of L 2( under the operator I , i.e. S(N ) I (S(L 2( 63 . Similarly, we have for R that S(R ) I (S(L 2( 11 . Proof: If f = I v with v 2 S(L 2( 13 then, by de nition of the native space norm, kfk = kvk 2 . The same holds in the second case. 2 The results of Pinkus book [11], in particular, Corollary 2.6 of Chapter IV yield: Theorem 11 Let be a positive Hilbert Schmidt kernel on L 2( with (15) Then, the n widths for the unit ball in N and R are given by dn (S(N ) L 2( 8 = dn (S(R ) L 2( 8 = n 1 ; respectively. In both cases, the subspace n : span ....

A. Pinkus. n{widths in Approximation Theory. Springer, 1985.

....of the results to the spaces with the derivatives bounded in L p norms. 1. Introduction In this paper, we derive a relatively simple to use, piecewise constant algorithm for approximating functions in a weighted L 1 sense. Function approximation has been studied quite extensively, see e.g. [3, 4, 6, 7] and the papers cited there. However, such problems were considered mainly for functions with a bounded domain D, say . The worst case complexity of weighted approximation over unbounded domains D has recently been studied in, e.g. 2, 9] assuming that the corresponding function classes ....

A. Pinkus, n-Widths in Approximation Theory, Springer-Verlag, Berlin, 1985.

....in the quantity sup g # F L g, d, q which is usually referred to as the approximation error of F by H . The larger d, the richer the class H , and the lower the error sup g # F L g, d, q The classical field of approximation theory (cf. Lorentz, Golitschek, and, Makovoz [16] Pinkus [21]) has many established results on the estimation of this error for numerous combinations of target classes F and hypothesis classes H . In this paper we consider another alternative for controlling L g, d, q which is conceptually different from the one above. Instead of enriching the learner ....

A. Pinkus, n-Widths in Approximation Theory," Springer-Verlag, New York, 1985.

....of a matrix A is the first singular vector of the matrix A multiplied by the square root of the corresponding singular value. Calling (U; S; V ) the singular value decomposition of A, and U the columns of U and oe i = S i;i the singular values of A we have: p = oe 1 U (3) Proof : See [6]. Notice that A = A and therefore U = V . Also: since A = A p is also equal to the eigenvector v 1 of A with largest eigenvalue i : p = 1=2 1 v 1 . Let s take a look at Fig. 2 where the function p that we obtain for the example of Fig. 1 is shown. We may notice that p takes values ....

A. Pinkus. n-Widths in Approximation Theory. Springer Verlag, 1985.

....all i 2 N we actually have a C i . For each C i we denote by H i the best approximating j plane, which is de ned in the following way: For every j plane H let (H) denote the minimal number such that C i H B d . Then H i is the plane for which (H) becomes minimal. It is known (see [P]) that we have (H i ) j 2)r j 1 (conv C i ) and thus (H i ) is bounded. By possibly translating C i by lattice vectors t i and taking a suitable subsequence we may assume that H i H, where convergence of planes is de ned in an obvious way, and that (H i ) As it is easier to ....

A. Pinkus: n{Widths in Approximation Theory, Springer{Verlag, Berlin, 1985.

....framework. The development of this paper follows the philosophical principles of George Zames on an information based theory of feedbackcontrol, identification, and adaptation [12] Complexity issues in modeling and identification have been pursued bymany researchers, including [11] 7] 5][4] [8] 6] The concept of complexityintroduced in this paper is an extension of the Kolmogorov entropy[3] 11] which characterizes the complexity of a set of functions by the minimal number of functions which can approximate the set to a precision level . While the Kolmogoroventropy has been ....

A. Pinkus, n-widths in Approximation Theory, Springer-Verlag, 1985.

....investigation of such problems and for an exposition of results. Even some of these results have been rediscovered. So called Karhunen Love method (with a different name) is known and widely used also in classical approximation theory and functional analysis (cf. Schmidt (1907) Ismagilov (1968) Pinkus (1985)) Further, for Gaussian processes, approximation problems in uniform metrics are closely related to extreme value theory (cf. Leadbetter, Lindgren, and Rootzn (1983) Piterbarg (1995) Many problems in time and or memory consuming computer experiments deal with approximation errors and optimal ....

....(1936) allows, in particular, to compare different methods in conventional approximation theory. This approach is based on n widths of classes of functions and gives an approximation rate which is optimal in some sense for certain classes of smooth functions (see, e.g. Tikhomirov (1974) Pinkus (1985)) We apply a similar method for random processes. Several types of extremal characteristics of an approximation accuracy are proposed. For Hlder s class of random processes, the exact orders of these characteristics are found in different metrics. This paper is organized as follows. In Section 2, ....

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Pinkus, A. (1985). n-Widths in Approximation Theory. Springer-Verlag, Berlin.

.... theory, application of such methods shows that some sets of functions defined by smoothness conditions exhibit the curse of dimensionality: the approximants converge at rate O( 1 d p n ) where d is the number of variables and n is the dimension of the approximating linear space (see, e.g. Pinkus, 1986). Our results show that these arguments are not applicable to approximation by Heaviside perceptron networks. 5 Note that the results from Section 3 cannot be extended to perceptron networks with differentiable activation functions, e.g. the logistic sigmoid or hyperbolic tangent. For such ....

Pinkus, A. (1986). n-Width in Approximation Theory . Berlin: Springer.

....(information) complexity comp( R) is equal to the minimal number of function and derivative evaluations needed to obtain error . We stress that the parameters p and q are not related. Let fl = r 1=q Gamma 1=p be positive, and let s = fl for p q, and s = r for p q. It is known, see e.g. [5, 6], that comp( R) Theta R fl 1=s with the factors in the Theta notation 1 independent of and R. Hence the complexity goes to infinity with R. In this paper, we study approximation of smooth univariate functions defined over the unbounded domain of reals, IR = Gamma1; 1) ....

.... factor c 2 [1; 2] we have r(n; R) c inf card(N) n sup fkfk q : f 2 F ; N (f) 0g : 2) By a standard change of variables one can verify that r(n; R) R fl r(n; 1) with fl = r 1=q Gamma 1=p: 3) Moreover, there are many results establishing the behavior of r(n; 1) see, e.g. [5, 6]. It equals infinity when n r Gamma 1 and, for n r, it is proportional to r(n; 1) Theta i n Gammas j with s = fl if p q; r if p q: 4) Observe that s 0, and s = 0 iff fl = 0. The latter holds iff r = 1; p = 1; q = 1. We need to guarantee that r(n; 1) tends to zero as n goes to ....

A. Pinkus, n-Widths in Approximation Theory, Springer-Verlag, Berlin, 1985.

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A. Pinkus, n-Widths in Approximation Theory, Springer-Verlag, Berlin, 1985.

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A. Pinkus, n-Widths in Approximation Theory, Springer-Verlag, Berlin, 1985.

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Pinkus, A., n-Widths in Approximation Theory, Springer-Verlag, Berlin, 1985.

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A. Pinkus, n-Widths in Approximation Theory," Springer-Verlag, Berlin, 1985.

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A. Pinkus, n-Widths in Approximation Theory, Springer (1985),

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A. Pinkus. n-Widths in Approximation Theory. Springer Verlag, 1985.

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A. Pinkus, n-Widths in Approximation Theory. New York: Springer, 1983.

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Pinkus, A., n-Width in Approximation Theory, Springer-Verlag, New York, 1985.

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A. Pinkus, n-widths in Approximation Theory, Springer-Verlag, 1982.

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A. Pinkus. n-Widths in Approximation Theory. Springer Verlag, 1985.

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A. Pinkus. n-Widths in Approximation Theory. Springer-Verlag, Berlin, 1985.

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A. Pinkus, N-widths in Approximation Theory (Springer, New York, 1986).

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A. Pinkus, n{Widths in Approximation Theory, Springer-Verlag, Berlin, 1985.

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Pinkus, A. n-widths in Approximation Theory. Springer-Verlag, 1985.

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A. Pinkus (1985): n-widths in approximation theory. Springer.

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