Journal ArticleDOI
Constructive Approximation by Superposition of Sigmoidal Functions
Danilo Costarelli,Renato Spigler +1 more
TLDR
In this paper, a constructive theory for approximating func- tions of one or more variables by superposition of sigmoidal functions is developed, which is done in the uniform norm as well as in the L p norm.Abstract:
In this paper, a constructive theory is developed for approximating func- tions of one or more variables by superposition of sigmoidal functions. This is done in the uniform norm as well as in the L p norm. Results for the simultaneous approx- imation, with the same order of accuracy, of a function and its derivatives (whenever these exist), are obtained. The relation with neural networks and radial basis func- tions approximations is discussed. Numerical examples are given for the purpose of illustration.read more
Citations
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Understanding deep learning requires rethinking generalization
TL;DR: The authors showed that deep neural networks can fit a random labeling of the training data, and that this phenomenon is qualitatively unaffected by explicit regularization, and occurs even if the true images are replaced by completely unstructured random noise.
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Machine Learning Force Fields
Oliver T. Unke,Stefan Chmiela,Huziel E. Sauceda,Michael Gastegger,Igor Poltavsky,Kristof T. Schütt,Alexandre Tkatchenko,Klaus-Robert Müller +7 more
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Controllable Invariance through Adversarial Feature Learning
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Journal ArticleDOI
Approximation results for neural network operators activated by sigmoidal functions
Danilo Costarelli,Renato Spigler +1 more
TL;DR: This paper studies pointwise and uniform convergence, as well as the order of approximation, for a family of linear positive neural network operators activated by certain sigmoidal functions, and shows that for C(1)-functions, the orders can be generalized to handle multivariate functions as well.
References
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Journal ArticleDOI
Approximation by superpositions of a sigmoidal function
TL;DR: It is demonstrated that finite linear combinations of compositions of a fixed, univariate function and a set of affine functionals can uniformly approximate any continuous function ofn real variables with support in the unit hypercube.
Book
Functional Analysis, Sobolev Spaces and Partial Differential Equations
TL;DR: In this article, the theory of conjugate convex functions is introduced, and the Hahn-Banach Theorem and the closed graph theorem are discussed, as well as the variations of boundary value problems in one dimension.
Journal ArticleDOI
Universal approximation using radial-basis-function networks
Jooyoung Park,Irwin W. Sandberg +1 more
TL;DR: It is proved thatRBF networks having one hidden layer are capable of universal approximation, and a certain class of RBF networks with the same smoothing factor in each kernel node is broad enough for universal approximation.
Journal ArticleDOI
Universal approximation bounds for superpositions of a sigmoidal function
TL;DR: The approximation rate and the parsimony of the parameterization of the networks are shown to be advantageous in high-dimensional settings and the integrated squared approximation error cannot be made smaller than order 1/n/sup 2/d/ uniformly for functions satisfying the same smoothness assumption.
Book
Mathematical Models in Population Biology and Epidemiology
TL;DR: This paper presents a series of models for continuous single-species and multi-species population models, and a model forStructured Population Models, which combines continuous and discrete models for populations with spatial distribution.