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Vector Algebra: Method of Components01:08

Vector Algebra: Method of Components

It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like addition, subtraction, and multiplication. There are two ways to circumvent this algebraic complexity. One way is to draw the vectors to scale, as in navigation, and read approximate vector lengths and angles (directions) from the graphs. The other way is to use the method of components.
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The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
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Updated: May 31, 2026

Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases
09:33

Diffusion Tensor Magnetic Resonance Imaging in the Analysis of Neurodegenerative Diseases

Published on: July 28, 2013

Discriminant locally linear embedding with high-order tensor data.

Xuelong Li1, Stephen Lin, Shuicheng Yan

  • 1School of Computer Science and Information Systems, Birkbeck College, University of London, London, UK.

IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics : a Publication of the IEEE Systems, Man, and Cybernetics Society
|March 20, 2008
PubMed
Summary
This summary is machine-generated.

Discriminant Locally Linear Embedding (DLLE) unifies dimensionality reduction methods. DLLE and its variants, including linear (DLLE/L) and tensor versions, enhance classification accuracy in gait and face recognition tasks.

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Area of Science:

  • Computer Science
  • Machine Learning
  • Pattern Recognition

Background:

  • Traditional dimensionality reduction algorithms lack a unified framework.
  • The out-of-sample problem remains a challenge in visual recognition.

Purpose of the Study:

  • To propose Discriminant Locally Linear Embedding (DLLE), a novel manifold learning technique.
  • To address the out-of-sample problem using linear (DLLE/L) and tensorized (DLLE) versions of DLLE.
  • To enhance classification accuracy in gait and face recognition.

Main Methods:

  • Developed DLLE by preserving local geometry within classes and maximizing inter-class separability.
  • Proposed DLLE/L for vector data and tensorized DLLE for high-order tensor input.
  • Applied DLLE-based methods to gait and face recognition tasks.

Main Results:

  • DLLE, DLLE/L, and tensorized DLLE outperformed related linear discriminant analysis methods.
  • DLLE/L showed greater effectiveness than linearized Locally Linear Embedding (LLE).
  • Tensor-based algorithms surpassed linear algorithms for high-order data.
  • DLLE/L achieved higher accuracy in gait recognition than state-of-the-art methods.

Conclusions:

  • DLLE provides a unified framework for dimensionality reduction.
  • DLLE variants effectively address the out-of-sample problem in visual recognition.
  • DLLE-based approaches offer superior performance, particularly for gait recognition.