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

Partial Fractions01:28

Partial Fractions

107
A partial fraction is a component of a rational expression represented as the sum of simpler fractions. When a rational function is expressed as a ratio of two polynomials, it can often be decomposed into a sum of fractions whose denominators are simpler polynomials, typically linear or irreducible quadratic factors. This process is called partial fraction decomposition, and it is used to simplify complex expressions for integration, solving equations, or analysis.Partial fraction decomposition...
107
Quadratic Equations01:29

Quadratic Equations

162
A quadratic equation is an algebraic expression where a variable is raised to the second power and combined with its first power and a constant; all equated to zero. These equations are frequently used to model relationships involving area, motion, and optimization. The general representation of a quadratic equation iswhere a, b, and c are real values, and a is nonzero to ensure the presence of the squared term.One method for solving a quadratic equation involves rewriting it as a product of...
162
Quadratic Equations in the Complex Number System01:29

Quadratic Equations in the Complex Number System

169
A quadratic equation in the form ax2+bx+c=0 can have solutions that vary in nature depending on the value of the discriminant, b2−4ac. In this expression, a is the coefficient of the quadratic term x2, b is the coefficient of the linear term x, and c is the constant term. When the discriminant is negative, the equation has no real number solutions. However, by introducing complex numbers through the imaginary unit i, defined by i=-1, these equations can still be solved.The square root of...
169
Quadratic Models01:23

Quadratic Models

97
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...
97
Fundamental Theorem of Algebra01:30

Fundamental Theorem of Algebra

120
The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as:  with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the...
120
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

206
Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
206

You might also read

Related Articles

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

Sort by
Same author

Blood-Activating and Stasis-Removing Chinese Patent Medicine in Perioperative Period of Percutaneous Coronary Intervention for Myocardial Infarction: A Systematic Review and Bayesian Network Meta-Analysis of Randomized Controlled Trials.

Journal of evidence-based medicine·2026
Same author

Antidepressant use and dementia, cognitive measures, and neuroimaging outcomes: A population-based cohort study.

Psychological medicine·2026
Same author

Single-cell transcriptomic analysis identifies a tumor-enriched GIMAP7⁺ CD4⁺ naïve T cell population with a unique immunoregulatory state in lung adenocarcinoma.

Functional & integrative genomics·2026
Same author

Vitamin D Supplementation in Children with Asthma: An Umbrella Review.

Nutrients·2026
Same author

Stochastic approximation to contrastive learning.

Neural networks : the official journal of the International Neural Network Society·2026
Same author

Clustering of multimorbidity in stroke and transient ischaemic attack survivors: a population-based study.

BMC medicine·2026

Related Experiment Video

Updated: Dec 12, 2025

Author Spotlight: Integrated Multi-Omics Analysis for Unveiling Multicellular Immune Signatures in Clinical Heart Attack Cohorts
08:51

Author Spotlight: Integrated Multi-Omics Analysis for Unveiling Multicellular Immune Signatures in Clinical Heart Attack Cohorts

Published on: September 20, 2024

1.9K

Unified development of multiplicative algorithms for linear and quadratic nonnegative matrix factorization.

Zhirong Yang1, Erkki Oja

  • 1Department of Information and Computer Science, Aalto University, Aalto FI-00076, Finland. zhirong.yang@aalto.fi

IEEE Transactions on Neural Networks
|October 20, 2011
PubMed
Summary

This study introduces a unified method for developing convergent multiplicative algorithms in nonnegative matrix factorization (NMF). The approach ensures guaranteed objective function decrease for various dissimilarity measures, improving optimization reliability.

More Related Videos

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
Using Cholesky Decomposition to Explore Individual Differences in Longitudinal Relations between Reading Skills
06:52

Using Cholesky Decomposition to Explore Individual Differences in Longitudinal Relations between Reading Skills

Published on: September 17, 2019

6.7K

Related Experiment Videos

Last Updated: Dec 12, 2025

Author Spotlight: Integrated Multi-Omics Analysis for Unveiling Multicellular Immune Signatures in Clinical Heart Attack Cohorts
08:51

Author Spotlight: Integrated Multi-Omics Analysis for Unveiling Multicellular Immune Signatures in Clinical Heart Attack Cohorts

Published on: September 20, 2024

1.9K
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
Using Cholesky Decomposition to Explore Individual Differences in Longitudinal Relations between Reading Skills
06:52

Using Cholesky Decomposition to Explore Individual Differences in Longitudinal Relations between Reading Skills

Published on: September 17, 2019

6.7K

Area of Science:

  • Machine Learning
  • Numerical Optimization
  • Data Analysis

Background:

  • Multiplicative updates are common in nonnegative matrix factorization (NMF) but lack general convergence proofs for diverse dissimilarity measures.
  • Existing convergence proofs rely on specific auxiliary functions, limiting applicability.

Purpose of the Study:

  • To develop a general approach for deriving convergent multiplicative algorithms for NMF.
  • To extend these algorithms to nonseparable cases and second-order factorizations.
  • To ensure theoretical guarantees of objective function decrease and practical optimality.

Main Methods:

  • A general method for deriving auxiliary upper-bounding functions for NMF problems with monomial objectives.
  • Extension of the method to nonseparable divergences like gamma and Renyi divergence.
  • Development of multiplicative algorithms for second-order approximative factorizations in NMF.

Main Results:

  • The proposed unified principle successfully derives convergent multiplicative algorithms for various NMF dissimilarity measures, including alpha- and beta-divergences.
  • The method is extended to handle nonseparable cases and second-order factorizations.
  • Numerical experiments show satisfactory Karush-Kuhn-Tucker optimality and immunity to descent violations common in conventional methods.

Conclusions:

  • The developed unified approach provides a robust framework for creating convergent multiplicative NMF algorithms.
  • This significantly advances NMF optimization by offering broader applicability and guaranteed performance.
  • The new algorithms demonstrate superior reliability and optimality compared to traditional methods.