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Linear Approximations

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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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...

Linearization and Approximation

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Linear Approximation in Frequency Domain

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Cluster Sampling Method

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Related Experiment Video

Updated: Jun 17, 2026

Image Recognition and Parameter Analysis of Concrete Vibration State Based on Support Vector Machine

Image Recognition and Parameter Analysis of Concrete Vibration State Based on Support Vector Machine

Published on: January 5, 2024

Scaling up support vector machines using nearest neighbor condensation.

Fabrizio Angiulli¹, Annabella Astorino

¹Department of Electronic, Computer Science and Systems Engineering, University of Calabria, Rende (CS), Italy. f.angiulli@deis.unical.it

IEEE Transactions on Neural Networks

|January 15, 2010

Summary

This summary is machine-generated.

The Fast Condensed Nearest Neighbor Support Vector Machine (FCNN-SVM) classifier significantly speeds up Support Vector Machines (SVMs) for large datasets. This approach drastically reduces training and testing times with only a minor accuracy trade-off.

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Image Recognition and Parameter Analysis of Concrete Vibration State Based on Support Vector Machine

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Published on: January 5, 2024

Visualization Method for Proprioceptive Drift on a 2D Plane Using Support Vector Machine

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Published on: October 27, 2016

Area of Science:

Machine Learning
Data Mining
Pattern Recognition

Background:

Support Vector Machines (SVMs) are powerful classification algorithms but struggle with large, multidimensional datasets due to computational complexity.
The efficiency of SVMs is often limited by the number of support vectors (SVs) and the time required for training and testing.

Purpose of the Study:

To introduce and evaluate the Fast Condensed Nearest Neighbor Support Vector Machine (FCNN-SVM) classifier.
To demonstrate the practical applicability of SVMs on large-scale data collections.

Main Methods:

The FCNN-SVM classifier integrates the Fast Condensed Nearest Neighbor (FCNN) condensation rule with the Support Vector Machine (SVM) algorithm.
Experimental analysis was conducted on very large and multidimensional datasets to compare FCNN-SVM with standard SVM.

Main Results:

FCNN-SVM achieved a speed improvement of one to two orders of magnitude compared to standard SVM on large datasets.
The number of support vectors (SVs) was more than halved when using FCNN-SVM, leading to drastic reductions in training and testing times.
A slight decrease in classification accuracy was observed as a trade-off for the significant speed gains.

Conclusions:

FCNN-SVM offers a viable and efficient alternative to standard SVM for applications demanding rapid response times.
The method effectively addresses the scalability challenges of SVMs in processing large and complex datasets.
This approach enhances the practicality of SVMs in real-world scenarios where computational efficiency is critical.