Related Experiment Video
Updated: Jun 16, 2026
13:44
Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
Published on: August 30, 2013
Generalized low-rank approximations of matrices revisited
Jun Liu1, Songcan Chen, Zhi-Hua Zhou
1Department of Computer Science and Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China. j.liu@nuaa.edu.cn
IEEE Transactions on Neural Networks
|February 23, 2010
Summary
Generalized low-rank approximations of matrices (GLRAM) offer advantages over singular value decomposition (SVD) in computation time and compression. This study reveals GLRAM
Area of Science:
- Numerical Analysis
- Machine Learning
- Data Science
Background:
- Generalized Low-Rank Matrix Approximation (GLRAM) is an efficient alternative to Singular Value Decomposition (SVD).
- GLRAM offers benefits in computation time, compression ratio, and classification performance.
- Existing literature lacks exploration of fundamental properties and solutions to crucial problems in GLRAM.
Purpose of the Study:
- To revisit and explore fundamental properties of GLRAM.
- To establish the relationship between GLRAM and SVD.
- To address open problems concerning GLRAM's objective function and compression performance.
Main Methods:
- Comparative analysis of GLRAM and SVD objective functions.
- Derivation of a lower bound for the GLRAM objective function.
- Theoretical analysis of GLRAM's compression performance based on minimizing the lower bound.
Main Results:
- GLRAM's objective function is identical to SVD's, differing only in constraints.
- A lower bound for the GLRAM objective function is derived, with conditions for achieving it.
- Theoretical justification is provided for optimal GLRAM reconstruction error when left and right transformations have equal columns.
Conclusions:
- GLRAM shares a close relationship with SVD, differing primarily in constraints.
- The study provides theoretical underpinnings for GLRAM's effectiveness in compression.
- This work addresses open questions, enhancing the understanding and usability of GLRAM.
Related Concept Videos
01:29Application of Linearization and Approximation
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...
01:23Linear 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...
01:30Gaussian Elimination: Problem Solving
Systems of linear equations in several variables are pivotal in modeling complex scenarios involving multiple unknowns and constraints. Such systems are widely used in various fields to represent relationships where several conditions must be simultaneously satisfied. Each variable in the system corresponds to an unknown quantity, while each equation imposes a linear constraint, leading to a structured approach for analyzing and solving real-world problems.A system of three equations with three...
01:26Linearization and Approximation
Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
01:21Linear Approximation in Time Domain
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, the...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
01:19Routh-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...
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...