Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Residuals and Least-Squares Property01:11

Residuals and Least-Squares Property

8.8K
The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative, and the line overestimates the actual data value for y.
The process of fitting the best-fit...
8.8K
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

302
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....
302
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

269
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,...
269
Vector Algebra: Method of Components01:08

Vector Algebra: Method of Components

18.5K
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.
In many applications, the magnitudes and directions of...
18.5K
Linear time-invariant Systems01:23

Linear time-invariant Systems

791
A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
791
Quadratic Models01:23

Quadratic Models

117
Quadratic models are mathematical representations used to describe relationships in which the rate of change changes at a constant rate. These models appear in a wide variety of natural and engineered systems, especially those involving motion, forces, and optimization. One common application is analyzing the vertical motion of objects influenced by gravity, such as a ball thrown into the air.In such scenarios, the object's height changes over time in a curved pattern, rising to a maximum point...
117

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Kernel Stein Discrepancy on Lie Groups: Theory and Applications.

IEEE transactions on information theory·2024
Same author

HIGHER ORDER GAUGE EQUIVARIANT CONVOLUTIONS FOR NEURODEGENERATIVE DISORDER CLASSIFICATION.

Proceedings. IEEE International Symposium on Biomedical Imaging·2024
Same author

Horospherical Decision Boundaries for Large Margin Classification in Hyperbolic Space.

Advances in neural information processing systems·2024
Same author

An Empirical Bayes Approach to Shrinkage Estimation on the Manifold of Symmetric Positive-Definite Matrices.

Journal of the American Statistical Association·2024
Same author

Geometric Deep Learning for Unsupervised Registration of Diffusion Magnetic Resonance Images.

Information processing in medical imaging : proceedings of the ... conference·2024
Same author

Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design.

Proceedings. IEEE Computer Society Conference on Computer Vision and Pattern Recognition·2023
Same journal

HardFlow: Hard-Constrained Sampling for Flow-Matching Models Via Trajectory Optimization.

IEEE transactions on pattern analysis and machine intelligence·2026
Same journal

Industrial Brain: Self-Evolving Neuro-Symbolic Autonomy with Causal Resilience for Cyber-Physical Systems.

IEEE transactions on pattern analysis and machine intelligence·2026
Same journal

Adaptive Hardness-Driven Dictionary Distillation for Incomplete Streaming View Clustering.

IEEE transactions on pattern analysis and machine intelligence·2026
Same journal

Mixture of Global and Local Experts with Diffusion Transformer for Controllable Face Generation.

IEEE transactions on pattern analysis and machine intelligence·2026
Same journal

Task-KV: Task-aware KV Cache Optimization via Semantic Differentiation of Attention Heads.

IEEE transactions on pattern analysis and machine intelligence·2026
Same journal

Achieving Text-based Person Retrieval with Any Granularity.

IEEE transactions on pattern analysis and machine intelligence·2026
See all related articles

Related Experiment Video

Updated: Dec 22, 2025

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

2.0K

Intrinsic Grassmann Averages for Online Linear, Robust and Nonlinear Subspace Learning.

Rudrasis Chakraborty, Liu Yang, Soren Hauberg

    IEEE Transactions on Pattern Analysis and Machine Intelligence
    |May 10, 2020
    PubMed
    Summary
    This summary is machine-generated.

    This study introduces a geometric framework for dimensionality reduction using principal component analysis (PCA) and Kernel PCA (KPCA). The new method offers a faster, robust, and online approach to KPCA, improving computational efficiency.

    More Related Videos

    Visualization Method for Proprioceptive Drift on a 2D Plane Using Support Vector Machine
    07:05

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

    Published on: October 27, 2016

    9.5K
    Deep Neural Networks for Image-Based Dietary Assessment
    13:19

    Deep Neural Networks for Image-Based Dietary Assessment

    Published on: March 13, 2021

    9.8K

    Related Experiment Videos

    Last Updated: Dec 22, 2025

    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

    2.0K
    Visualization Method for Proprioceptive Drift on a 2D Plane Using Support Vector Machine
    07:05

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

    Published on: October 27, 2016

    9.5K
    Deep Neural Networks for Image-Based Dietary Assessment
    13:19

    Deep Neural Networks for Image-Based Dietary Assessment

    Published on: March 13, 2021

    9.8K

    Area of Science:

    • Machine Learning
    • Computational Geometry
    • Data Science

    Background:

    • Principal Component Analysis (PCA) and Kernel PCA (KPCA) are essential for dimensionality reduction.
    • KPCA operates in infinite-dimensional spaces (reproducing Kernel Hilbert spaces - RKHS).
    • Existing methods can be computationally intensive.

    Purpose of the Study:

    • To develop a geometric framework for dimensionality reduction.
    • To compute principal linear subspaces in finite and infinite dimensions, including robust PCA.
    • To create a faster and online version of KPCA.

    Main Methods:

    • Utilizing a geometric framework based on the Grassmann manifold.
    • Computing the intrinsic average of subspaces spanned by observations.
    • Developing an efficient algorithm for projection onto the average subspace.

    Main Results:

    • The geometric framework successfully computes principal components and extends to RKHS.
    • A novel, efficient algorithm akin to KPCA is developed, offering substantial speed improvements.
    • A new online version of KPCA is introduced.

    Conclusions:

    • The proposed geometric approach provides a faster and more efficient alternative to traditional KPCA.
    • The developed algorithms demonstrate competitive performance on diverse datasets.
    • This work offers significant advancements in dimensionality reduction techniques.