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Related Concept Videos

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.
Calibration Curves: Linear Least Squares01:20

Calibration Curves: Linear Least Squares

A calibration curve is a plot of the instrument's response against a series of known concentrations of a substance. This curve is used to set the instrument response levels, using the substance and its concentrations as standards. Alternatively, or additionally, an equation is fitted to the calibration curve plot and subsequently used to calculate the unknown concentrations of other samples reliably.
For data that follow a straight line, the standard method for fitting is the 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 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...
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...

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Related Experiment Video

Updated: Jul 7, 2026

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator
06:45

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

A local linearized least squares algorithm for training feedforward neural networks.

O Stan1, E Kamen

  • 1School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250, USA.

IEEE Transactions on Neural Networks
|February 6, 2008
PubMed
Summary

A new Local Linearized Least Squares (LLLS) algorithm improves neural network training. LLLS offers better convergence than MEKA and DEKF, approaching GEKF performance without high computational cost.

Related Experiment Videos

Last Updated: Jul 7, 2026

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator
06:45

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

Area of Science:

  • Artificial Intelligence
  • Machine Learning
  • Computational Neuroscience

Background:

  • Global Extended Kalman Filter (GEKF) offers superior neural network training convergence and solution quality compared to gradient descent with error backpropagation.
  • GEKF's high computational intensity necessitates efficient alternatives like MEKA and DEKF, which use dimensional reduction or partitioning.

Purpose of the Study:

  • Introduce a novel training algorithm, Local Linearized Least Squares (LLLS), for feedforward neural networks.
  • Evaluate LLLS performance against existing efficient algorithms (MEKA, DEKF) and the benchmark GEKF.

Main Methods:

  • LLLS treats neuron-level system identification as recursive linearized least squares problems.
  • The objective function minimizes the sum of squared linearized backpropagated errors for each neuron.

Main Results:

  • LLLS demonstrated superior convergence on three benchmark problems compared to MEKA.
  • For highly coupled applications, LLLS outperformed DEKF.
  • The performance of LLLS closely matched that of GEKF in experimental evaluations.

Conclusions:

  • LLLS presents an efficient and effective alternative for training feedforward neural networks.
  • The algorithm balances computational efficiency with high-quality training outcomes, approaching GEKF's effectiveness.