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Nonlinear Approximation
 AMER. J. MATH
, 2006
"... We prove that if X is any 2regular scheme (in the sense of CastelnuovoMumford) then X is small. This means that if L is a linear space and Y := L X is finite, then Y is linearly independent in the sense that the dimension of the linear span of Y is deg Y 1. The converse is true and wellk ..."
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We prove that if X is any 2regular scheme (in the sense of CastelnuovoMumford) then X is small. This means that if L is a linear space and Y := L X is finite, then Y is linearly independent in the sense that the dimension of the linear span of Y is deg Y 1. The converse is true and wellknown for finite schemes, but false in general. The main result of this paper is that the converse, "small implies 2regular", is also true for reduced schemes (algebraic sets). This is proven by means of a delicate geometric analysis, leading to a complete classification: we show that the components of a small algebraic set are varieties of minimal degree, meeting in a particularly simple way. From the classification one can show that if X is 2regular, then so is X red , and so also is the projection of X from any point of X. Our result
Adaptive Nonlinear Approximations
, 1994
"... The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NPhard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing the dictionary waveforms which best matc ..."
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Cited by 73 (1 self)
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The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NPhard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing the dictionary waveforms which best
Nonlinear Approximation with Walsh Atoms
 Surface Fitting and Multiresolution Methods
, 1997
"... . As a model for nonlinear approximation from a redundant set of timefrequency atoms, we consider approximation in L 2 (IR) with linear combinations of Walsh at oms. Best approximation can be realized with a fast algorithm when the class of approximants is restricted to linear combinations of pai ..."
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Cited by 12 (3 self)
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. As a model for nonlinear approximation from a redundant set of timefrequency atoms, we consider approximation in L 2 (IR) with linear combinations of Walsh at oms. Best approximation can be realized with a fast algorithm when the class of approximants is restricted to linear combinations
Nonlinear approximation by trigonometric sums
 J. Fourier Anal. and Appl
, 1995
"... ABSTRACT. We investigate the Lperror of approximation to a function f ∈ Lp(Td) by a linear combination ∑ k ckek of n exponentials ek(x): = ei〈k,x 〉 = ei(k1x1+···+kd xd) d on T, where the frequencies k ∈ Zd are allowed to depend on f. We bound this error in terms of the smoothness and other propert ..."
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Cited by 23 (7 self)
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ABSTRACT. We investigate the Lperror of approximation to a function f ∈ Lp(Td) by a linear combination ∑ k ckek of n exponentials ek(x): = ei〈k,x 〉 = ei(k1x1+···+kd xd) d on T, where the frequencies k ∈ Zd are allowed to depend on f. We bound this error in terms of the smoothness and other
Approximation by Superpositions of a Sigmoidal Function
, 1989
"... In this paper we demonstrate that finite linear combinations of compositions of a fixed, univariate function and a set ofaffine functionals can uniformly approximate any continuous function of n real variables with support in the unit hypercube; only mild conditions are imposed on the univariate fun ..."
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Cited by 1248 (2 self)
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continuous sigmoidal nonlinearity. The paper discusses approximation properties of other possible types of nonlinearities that might be implemented by artificial neural networks.
On Nonuniqueness in Nonlinear ...Approximation
"... : It is shown that under weak assumptions nonlinear L 2 approximation problems generally have unbounded numbers of local best approximations. This includes the rational and the exponential family of approximating functions. In addition, for a certain class of approximating families, we construct f ..."
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: It is shown that under weak assumptions nonlinear L 2 approximation problems generally have unbounded numbers of local best approximations. This includes the rational and the exponential family of approximating functions. In addition, for a certain class of approximating families, we construct
Results 1  10
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13,738