Singular Value Decomposition

Singular Value Decomposition (SVD)

AI Terms

Singular Value Decomposition (SVD)

Definition: Singular Value Decomposition (SVD) is a matrix factorization technique used to decompose a matrix into its singular values and vectors.

Detailed Description:

SVD is a powerful tool for data analysis and dimensionality reduction. It can be used to:

Reduce dimensionality: Identify the most important features in a dataset.
Data visualization: Visualize high-dimensional data in a lower-dimensional space.
Collaborative filtering: Recommend items to users based on their preferences.
Image compression: Compress images by discarding less important singular values.

SVD decomposes a matrix A into three matrices:

A = UΣV^T

where:

U is an orthogonal matrix containing the left singular vectors.
Σ is a diagonal matrix containing the singular values.
V^T is the transpose of an orthogonal matrix containing the right singular vectors.

Related Terms:

FAQs:

What is the difference between SVD and PCA?
- SVD and PCA are closely related, but SVD provides a more general decomposition of a matrix.
How is SVD used in image compression?
- SVD can be used to identify the most important features in an image and discard less important information, resulting in a compressed image.
What are the challenges of using SVD?
- SVD can be computationally expensive for large matrices.

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