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Related Concept Videos

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...
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...
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.
Reducing Line Loss01:18

Reducing Line Loss

In a three-phase circuit, line loss is an indicator of energy dissipated as heat due to the resistance of transmission lines. To address this, incorporating transformers into the system—a step-up transformer at the source and a step-down transformer at the load—is a strategic solution. Two three-phase transformers are introduced to improve this.
With a step-up transformer at the source, the voltage is increased, thereby reducing the current in the transmission lines since power loss in...
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...

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Related Experiment Videos

Multilayer neural networks for reduced-rank approximation.

K I Diamantaras1, S Y Kung

  • 1Siemens Corp. Res. Inc., Princeton, NJ.

IEEE Transactions on Neural Networks
|January 1, 1994
PubMed
Summary

This study presents new methods for reduced-rank linear approximation and generalized eigenvalue problems using artificial neural networks. These approaches unify existing techniques and solve problems with non-invertible autocorrelation matrices.

Related Experiment Videos

Area of Science:

  • Machine Learning
  • Linear Algebra
  • Signal Processing

Background:

  • Traditional methods for linear approximation and eigenvalue problems often assume invertibility of the autocorrelation matrix, limiting their applicability.
  • Artificial neural networks offer potential for solving complex linear algebra problems but require specific architectures and training methods.

Purpose of the Study:

  • To develop a unified framework for reduced-rank linear approximation that relaxes the invertibility assumption.
  • To explore artificial neural network models for solving the generalized eigenvalue problem.
  • To apply these novel methods to system identification problems with non-invertible autocorrelation matrices.

Main Methods:

  • Formulation of the general reduced-rank linear approximation problem without the invertibility assumption.
  • Analysis of two-layer linear neural networks trained with least-squares error for generalized singular value decomposition.
  • Investigation of sequential and parallel artificial neural network models (including Lateral Orthogonalization Network - LON) for the generalized eigenvalue problem using extended deflation concepts.

Main Results:

  • The proposed framework unifies various methods like linear regression, Wiener filtering, SVD, and principal component analysis (PCA).
  • Two-layer linear neural networks with reduced hidden units extract generalized singular value decomposition components.
  • Sequential and parallel neural network models successfully extract generalized eigenvector components, addressing limitations of previous methods.

Conclusions:

  • The developed methods provide a unified approach to linear approximation and eigenvalue problems, applicable even when autocorrelation matrices are non-invertible.
  • Artificial neural networks, particularly the LON, offer efficient and flexible solutions for extracting generalized singular vectors and eigenvectors.
  • These findings enable the identification of systems with non-invertible autocorrelation matrices, expanding the scope of system identification techniques.