Representation of functions by superpositions of a step or sigmoid function and their applications to neural network theory

Ito, Yoshifusa

doi:10.1016/0893-6080(91)90075-g

Cited by 198 publications

(86 citation statements)

References 3 publications

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“…In this paper we derive integral formulas corresponding to one-hidden-layer Heaviside networks, extending and unifying results in [8], [4], [13], and [19]. Some of the ideas in this paper appeared in [16].…”

Section: Introductionmentioning

confidence: 79%

“…Ito [13] and Carroll and Dickinson [4] treated both the odd and the even case, basing their work on Helgason's book on the Radon Transform [11], and obtained a representation for C ∞ functions of rapid descent (Ito) and C ∞ functions of compact support (Carroll and Dickinson).…”

Section: Alternative Representationsmentioning

confidence: 99%

“…Such integral formulas have been used to show that output functions from one-hidden-layer neural networks with suitable φ and finitely many units are dense in various function spaces (see, e.g., Funahashi [8], Carroll and Dickinson [4], and Ito [13]). Integral representations have also been used to estimate how accuracy of approximation varies with the number of hidden units (see, e.g., Barron [1], Girosi and Anzellotti [10], and Kůrková, Kainen and Kreinovich [19]).…”

Section: Introductionmentioning

confidence: 99%

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Integral combinations of Heavisides

Kainen

Kůrková

Vogt

2010

Mathematische Nachrichten

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Abstract:A sufficiently smooth function of d variables that decays fast enough at infinity can be represented pointwise by an integral combination of Heaviside plane waves (i.e., characteristic functions of closed half-spaces). The weight function in such a representation depends on the derivatives of the represented function. The representation is proved here by elementary techniques with separate arguments for even and odd d, and unifies and extends various results in the literature. An outline of the paper follows. Section 2 reviews neural networks and establishes notation, Section 3 discusses Green's functions and Green's second identity, and Section 4 describes functions of controlled decay and states our main theorem. We consider the 1-dimensional case in Section 5, provide necessary lemmas in Section 6, and prove the main theorem in Section 7. Section 8 has extensions and refinements of our representation and its relation to known results. The paper ends with a brief discussion. Keywords Feedforward neural networksFeedforward neural networks compute functions determined by the type of units and their interconnections. Each computational unit depends on two vector variables (an input and a parameter ), and is given by a function φ : R p × R d → R, where p and d are the dimensions of the parameter and input space respectively and R denotes the set of real numbers.One-hidden-layer networks, with hidden units based on a fixed function φ and a single linear output unit, yield functions f :where n is the number of hidden units, and w i ∈ R are the output weights and a i ∈ R p the input parameters of the i-th unit for i = 1, ..., n.A perceptron is a computational unit based on a function of the form φ ((v, b)

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Section: Introductionmentioning

confidence: 79%

Section: Alternative Representationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Integral combinations of Heavisides

Kainen

Kůrková

Vogt

2010

Mathematische Nachrichten

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“…The ARMA model is used to model the linear part and the MFNN is used to model the nonlinear part. The use of the MFNN is motivated by its ability to model any nonlinear function to any desired accuracy [12][13][14]. Considering single-input single-output systems, the output of the ARMA model is given by …”

Section: Introductionmentioning

confidence: 99%

“…In this study, we have developed a two-step method for identification of the Wiener model. In the first step, a small signal which ensures linear perturbation of the nonlinear system is applied to identify the linear dynamics using the recursive least square (RLS) algorithm [12]. This method of identifying the linear part using small-signal analysis is proposed in [4].…”

Section: Introductionmentioning

confidence: 99%

Use of multilayer feedforward neural networks in identification and control of Wiener model

Al-Duwaish

Karim

Chandrasekar

1996

IEE Proceedings - Control Theory and Applications

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The problem of identification and control of a Wiener model is studied. The proposed identification model uses a hybrid model consisting of a linear autoregressive moving average model in cascade with a multilayer feedforward neural network. A twostep procedure is proposed to estimate the linear and nonlinear parts separately. Control of the Wiener model can be achieved by inserting the inverse of the static nonlinearity in the appropriate loop locations. Simulation results illustrate the performance of the proposed method.

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Dynamic convolutional neural network based e‐waste management and optimized collection planning

Latha

Kalaiselvi

Sankriti

et al. 2022

Concurrency and Computation

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Electronic waste, also known as e‐waste, refers to electrical or electronic devices that are discarded from households and workplaces. These used e‐wastes are meant to be renovated, reused, recycled, or disposed of, and the processing of these wastes often causes disease and harms the environment. As a result, it is important to handle waste and collect it from the disposal site on a regular basis. Besides, in order to separate precious metals from discarded waste, it is important to identify them by category. Therefore, this article proposes a novel method known as e‐waste management by exploiting the dynamic convolutional neural network (DCNN). This enhances the classification accuracy with the aid of exactly mapping the features of the images. Meanwhile, the collection of waste can be optimized in order to reduce the distance and time. The e‐wastes in the smart garbage bin are frequently monitored by smartphone applications to collect the waste on time. Moreover, it also significantly reduces the training error, classification error, localization error, and validation error on the test images. The experimental depicts that the proposed method hones up the classification accuracy to the great extent.

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Representation of functions by superpositions of a step or sigmoid function and their applications to neural network theory

Cited by 198 publications

References 3 publications

Integral combinations of Heavisides

Integral combinations of Heavisides

Use of multilayer feedforward neural networks in identification and control of Wiener model

Dynamic convolutional neural network based e‐waste management and optimized collection planning

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