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About 10,000 papers for “nonlinear approximation”

Nonlinear Approximation

D. Eisenbud, M. Green, K. Hulek et al. · 2006

We prove that if X is any 2-regular scheme (in the sense of Castelnuovo-Mumford) 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 well-known for finite schemes, but false in general. The main result of this paper is that the converse, "small implies 2-regular", 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 2-regular, then so is X red , and so also is the projection of X from any point of X. Our result

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Nonlinear Value Function Approximation

Michael Y. Hu · Apress eBooks · 2023

In this chapter, we will explore the use of nonlinear methods to approximate value functions in reinforcement learning. Value functions play a crucial role in reinforcement learning as they estimate the expected reward an agent can obtain in a given state and follow the policy afterwards. In the previous chapter, we discussed linear methods that rely on constructing a feature vector and computing a weighted combination of features as the state or state-action value. However, these methods have limitations when applied to complex and challenging domains. In such cases, a simple linear combination may fail to capture all the relevant features and information. Additionally, determining the appropriate features that capture the necessary information for the task at hand requires substantial domain knowledge and experience.

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On Nonuniqueness in Nonlinear ...-Approximation

Immo Diener · 2007

: 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 functions with three global best approximations. The results apply for instance to exponential and rational approximating families with one nonlinear parameter. Finally, we extend results of Spieß and Braess for the finiteness of the number of local best approximations by rational functions. 1. General nonlinear L 2 --approximation In this section we show that general nonlinear L 2 --approximation problems have unbounded numbers of local best approximations. This extends results of Wolfe [7] for the special case of ordinary rational functions. We consider the Hilbert space H := L 2 [\\Gamma1; 1]. Let S be an open subset of R M and A a twice Fr'echet-differentiable map from S to H . Thus, e...

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Nonlinear approximation using Blaschke polynomials

Daniel Van Vliet · 2007

This dissertation, entitled Nonlinear Approximation Using Blaschke Polynomials, is motivated by questions arising from Empirical Mode Decomposition (EMD). EMD is a signal processing method which decomposes input signals into components called intrinsic mode functions (IMFs). These IMFs often have the desirable property that the instantaneous frequency of their analytic signals is positive. However, this is not always the case.;The first two chapters are introductions to approximation in general, and Empirical Mode Decomposition, respectively.;The third chapter presents a characterization of which analytic signals have the property of non-negative instantaneous frequency. These 'analytic signals with non-negative instantaneous frequency' (ASNIFs) are described using Hardy spaces on the unit disc. Analogous results are also found for analytic signals which are boundary values fof elements of Hardy spaces on the half-plane.;The fourth chapter describes in general terms how one might construct an approximation method using ASNIFs, or possibly some restricted subclass of ASNIFs, as approximants.;The fifth chapter introduces a special set of ASNIFs: Blaschke polynomials. These are functions which are linear combinations of Blaschke products, which are a special kind of ASNIF. Blaschke polynomials are a natural extension of other classical approximating sets, and have some interesting properties of their own. Some of these properties are explored.;The sixth chapter contains an implementation of a signal decomposition method using Blaschke polynomials, and a discussion of the results.;The seventh chapter further explores the conclusions of this work, and lays out future research questions.

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