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Related Concept Videos

Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
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...
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
Linearization and Approximation01:26

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
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...

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Related Experiment Videos

Comments on "Approximation capability in C(R(n)) by multilayer feedforward networks and related problems".

G B Huang, H A Babri

    IEEE Transactions on Neural Networks
    |February 7, 2008
    PubMed
    Summary

    The boundedness of activation functions is not strictly necessary or sufficient for neural network function approximation in C(Rn). Instead, boundedness combined with unequal limits at infinities provides sufficient, though not essential, conditions.

    Related Experiment Videos

    Area of Science:

    • * Neural network approximation theory
    • * Function approximation in C(Rn) spaces

    Background:

    • * Chen et al. explored uniform approximation of functions in C(Rn) using feedforward neural networks.
    • * They identified the critical role of activation function boundedness and conjectured its necessity and sufficiency for approximation theorems.

    Discussion:

    • * This study refutes the conjecture by Chen et al., demonstrating that boundedness alone is neither necessary nor sufficient for uniform approximation in C(Rn).
    • * The findings highlight that while boundedness is important, it does not fully capture the requirements for approximation capabilities.

    Key Insights:

    • * The condition of boundedness for activation functions is not a necessary or sufficient requirement for uniform approximation in C(Rn).
    • * A combination of boundedness and unequal limits at infinities for activation functions is sufficient, but not necessary, for approximation.

    Outlook:

    • * Further research is needed to precisely define the necessary and sufficient conditions for activation functions in neural network approximation.
    • * Exploring alternative activation function properties could lead to more robust and efficient approximation capabilities.