Covers: theory of Singular Value Decomposition
Estimated time needed to finish: 15 minutes
Questions this item addresses:
• How is the best rank r approximation of a matrix obtained using SVD?
• What is the Eckart-Young-Mirsky Theorem?
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1. Expansion of SVD decomposition as the sum of rank one matrices.
2. The best rank one, two, ... or r approximation (w.r.t. Frobenius norm (*)). Recall that
1. Eckart-Young-Mirsky Theorem: Best rank r approximation of a matrix.
• Definition (*): Given an mxn real matrix A, the Frobenius norm of A is defined as
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Steve Brunton
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Singular Value Decomposition (SVD)

Contributors
Total time needed: ~2 hours
Objectives
Learn the basics of SVD and its applications in dimensionality reduction
Potential Use Cases
Educational use. Gives basic introduction to SVD and some of its applications
Who is This For ?
INTERMEDIATEAnyone interested in SVD with basic linear algebra background.
Click on each of the following annotated items to see details.
VIDEO 1. Overview of Singular Value Decomposition (SVD)
• What is SVD?
• What are the factors of SVD?
• How are the factors of SVD related to the data matrix?
13 minutes
BOOK_CHAPTER 2. The theory of SVD
• What is the relation between the SVD factors of a matrix and its four fundamental subspaces?
• How is SVD computed?
• How can SVD be used for image compression?
18 minutes
VIDEO 3. Dimensionality Reduction via SVD
• How is the best rank r approximation of a matrix obtained using SVD?
• What is the Eckart-Young-Mirsky Theorem?
15 minutes
VIDEO 4. Choosing the Optimal Rank for Truncating SVD
• How to choose truncation value in SVD?
12 minutes
ARTICLE 5. (Optional) Matrix Completion and Low-Rank SVD via Fast Alternating Least Squares
• How to efficiently estimate missing entries of large matrices?
30 minutes

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