Covers: theory of Singular Value Decomposition

- How is the best rank r approximation of a matrix obtained using SVD?
- What is the Eckart-Young-Mirsky Theorem?

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- Expansion of SVD decomposition as the sum of rank one matrices.
- The best rank one, two, ... or r approximation (w.r.t. Frobenius norm (*)). Recall that

- 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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Contributors

- 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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