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:

• Matrix Decomposition
• Dimensionality Reduction
• Principal Component Analysis (PCA)
• Linear Algebra

FAQs:

1. What is the difference between SVD and PCA?

   • SVD and PCA are closely related, but SVD provides a more general decomposition of a matrix.

2. 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.

3. What are the challenges of using SVD?

   • SVD can be computationally expensive for large matrices.