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Related Concept Videos

Fundamental Theorem of Algebra01:30

Fundamental Theorem of Algebra

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 Complete Factorization...
Complex Zeros01:29

Complex Zeros

Complex zeros are the solutions to polynomial equations that include imaginary numbers, specifically, numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit defined by i2=-1. These zeros satisfy the equation P(x) = 0, where P(x) is a polynomial with real or complex coefficients. Since the complex number system includes all real numbers, it provides a complete framework for analyzing all possible roots of a polynomial.Every polynomial of degree n≥1 can be...
Real Zeros of Polynomials01:27

Real Zeros of Polynomials

Polynomials are algebraic expressions of terms with variables raised to non-negative integer powers. A central aspect of analyzing polynomial functions is determining their real zeros—values of the variable for which the polynomial evaluates to zero. These values represent the x-intercepts of the polynomial’s graph.The Rational Zeros Theorem lists possible rational solutions for a polynomial equation with integer coefficients. If f(x)=anxn+....+a0​, then every rational zero is of the form p/q​,...
Introduction to Polynomial Functions01:26

Introduction to Polynomial Functions

Polynomial functions are fundamental elements in algebra and calculus, defined by expressions that combine variables and constants through addition, subtraction, and multiplication, with the variable raised to nonnegative integer exponents. A general polynomial function of degree n is given byWhere an ≠ 0. The term anxn is the leading term, and an is the leading coefficient, while a0 is referred to as the constant term.Characteristics and ClassificationPolynomials are categorized by their...
Synthetic Disvision of Polynomials01:28

Synthetic Disvision of Polynomials

Synthetic division is an efficient algorithmic approach for dividing a polynomial by a linear binomial of the form x - c, where c is a real number. This method is helpful due to its streamlined process, which avoids the more cumbersome steps involved in the traditional long division of polynomials. It simplifies computation and serves as a practical tool for evaluating polynomials and identifying their factors.To perform synthetic division, one begins by listing the coefficients of the...
Routh-Hurwitz Criterion II01:19

Routh-Hurwitz Criterion II

In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first column of the Routh...

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Related Experiment Video

Updated: Jul 4, 2026

Application of Unsupervised Multi-Omic Factor Analysis to Uncover Patterns of Variation and Molecular Processes Linked to Cardiovascular Disease
08:51

Application of Unsupervised Multi-Omic Factor Analysis to Uncover Patterns of Variation and Molecular Processes Linked to Cardiovascular Disease

Published on: September 20, 2024

Nonnegative matrix factorization in polynomial feature space.

Ioan Buciu1, Nikos Nikolaidis, Ioannis Pitas

  • 1Department of Informatics, Aristotle University of Thessaloniki, Thessloniki 54006, Greece. ibuciu@uoradea.ro

IEEE Transactions on Neural Networks
|June 11, 2008
PubMed
Summary

This study introduces a generalized nonnegative matrix factorization (NMF) algorithm using kernel functions in Hilbert space. This approach captures high-order dependencies for improved data decomposition and feature discovery.

Related Experiment Videos

Last Updated: Jul 4, 2026

Application of Unsupervised Multi-Omic Factor Analysis to Uncover Patterns of Variation and Molecular Processes Linked to Cardiovascular Disease
08:51

Application of Unsupervised Multi-Omic Factor Analysis to Uncover Patterns of Variation and Molecular Processes Linked to Cardiovascular Disease

Published on: September 20, 2024

Area of Science:

  • Machine Learning
  • Data Analysis
  • Computer Vision

Background:

  • Traditional methods like PCA, ICA, and FA are used for latent variable discovery.
  • Nonnegative Matrix Factorization (NMF) decomposes data into lower dimensions with nonnegativity constraints.
  • Standard NMF minimizes objective functions in Euclidean space using L(2)-norm.

Purpose of the Study:

  • To generalize the NMF algorithm by extending its objective function to a Hilbert space.
  • To incorporate kernel functions to capture high-order dependencies in data.
  • To maintain nonnegativity constraints on basis images and coefficients.

Main Methods:

  • Translating the NMF objective function into a Hilbert space.
  • Employing kernel functions to model complex relationships.
  • Minimizing the objective function under nonnegativity constraints.

Main Results:

  • Developed a generalized NMF approach capable of capturing high-order dependencies.
  • Successfully maintained nonnegativity constraints on both basis images and coefficients.
  • Demonstrated the approach's potential in facial expression and face recognition tasks.

Conclusions:

  • The proposed Hilbert space NMF with kernels offers a powerful generalization of standard NMF.
  • This method enhances feature discovery by modeling complex, high-order data dependencies.
  • The approach shows promise for applications in image analysis and recognition.