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Related Concept Videos

Approximate Integration01:24

Approximate Integration

In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
Application of Linearization and Approximation01:29

Application of Linearization and Approximation

A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during...
Linear Approximations01:23

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...
Accuracy, limits, and approximation01:28

Accuracy, limits, and approximation

Accuracy, limits, and approximations are common in many fields, especially in engineering calculations. These concepts are imperative for ensuring that a given value is as close as possible to its true value.
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Improper Integrals: Infinite Intervals01:29

Improper Integrals: Infinite Intervals

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Linearization and Approximation01:26

Linearization and Approximation

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Related Experiment Videos

An integral upper bound for neural network approximation.

Paul C Kainen1, Vera Kůrková

  • 1Department of Mathematics, Georgetown University, Washington, DC 20057-1233, USA. kainen@georgetown.edu

Neural Computation
|July 29, 2009
PubMed
Summary

This study analyzes the complexity of one-hidden-layer neural networks. Researchers derived bounds on approximation error for networks with many hidden units, improving understanding of their performance.

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Area of Science:

  • Artificial Intelligence
  • Machine Learning
  • Computational Theory

Background:

  • One-hidden-layer neural networks are fundamental in machine learning.
  • Understanding their approximation capabilities is crucial for algorithm development.
  • Existing theories often lack precise error bounds for increasing network complexity.

Purpose of the Study:

  • To investigate the approximation complexity of one-hidden-layer neural networks.
  • To derive upper bounds on the rate of approximation error decrease.
  • To apply these findings to perceptron networks.

Main Methods:

  • Utilizing tools from nonlinear approximation theory.
  • Employing integration theory, specifically Bochner integration.
  • Analyzing functions with suitable integral representations.

Main Results:

  • Established upper bounds on the speed of approximation error decrease.
  • Demonstrated how error diminishes as the number of network units increases.
  • Bounds are applicable across various function norms.

Conclusions:

  • The study provides theoretical insights into the approximation power of neural networks.
  • The derived bounds enhance the understanding of network complexity and performance.
  • Findings are relevant for the design and application of perceptron networks.