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Related Concept Videos

Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

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.
Linear Approximations01:23

Linear 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...
Linear Approximation in Time Domain01:21

Linear 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...
Linearization and Approximation01:26

Linearization 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...
Application of Linearization and Approximation01:29

Application 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...
Approximate Integration01:24

Approximate Integration

In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...

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Related Experiment Videos

Approximate nearest subspace search.

Ronen Basri1, Tal Hassner, Lihi Zelnik-Manor

  • 1Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel. ronen.basri@weizmann.ac.il

IEEE Transactions on Pattern Analysis and Machine Intelligence
|June 2, 2010
PubMed
Summary
This summary is machine-generated.

This study introduces a novel method for fast Approximate Nearest Subspace search in large databases. It efficiently finds similar subspaces, improving pattern recognition and machine vision applications.

Related Experiment Videos

Area of Science:

  • Computer Science
  • Machine Learning
  • Pattern Recognition

Background:

  • Subspace representations are widely used in pattern recognition, machine vision, and statistical learning.
  • Efficiently searching large subspace databases remains a significant challenge.
  • Existing methods lack comprehensive solutions for diverse query and database characteristics.

Purpose of the Study:

  • To develop a general and efficient solution for Approximate Nearest Subspace search.
  • To address challenges including varying dimensionalities and mixed subspace dimensions in databases.
  • To reduce the complex subspace search problem to a well-established point-based search.

Main Methods:

  • A novel mapping technique transforms subspaces into points.
  • This reduction enables the application of Approximate Nearest Neighbor search algorithms.
  • Theoretical proofs establish correctness and error bounds for the proposed mapping.

Main Results:

  • The method uniformly handles queries as points or subspaces of differing dimensions.
  • Experiments on synthetic and real data validate the approach's effectiveness.
  • Significantly faster search times were achieved compared to exact nearest subspace search.

Conclusions:

  • The proposed mapping provides an efficient solution for Approximate Nearest Subspace search.
  • This approach enhances the practicality of subspace methods in large-scale applications.
  • The technique offers a balance between search speed and accuracy.