# Lecture 22 – Dimensionality Reduction

by Josh Hug (Fall 2019)

**Important:** This lecture is a combination of two lectures from the Fall 2019 semester.

- There are a couple of small typos in 20.4. To check whether a set of vectors is an orthonormal set, we should check whether V^T @ V is the identity matrix (not V @ V^T). For matrices whose columns form an orthonormal set, the property that the matrix’s transpose is equivalent to its inverse only holds true if the matrix is square.
- There is a set of extra slides at the end of the lecture slides. These slides contain a review of concepts in linear algebra such as matrix multiplication and rank.

Video | Quick Check | |
---|---|---|

22.1 Dimensionality. Visualizing high-dimensional data. |
22.1 | |

22.2 More visualizations of high-dimensional data. |
22.2 | |

22.3 Matrix decomposition, redundancy, and rank. Introduction to the singular value decomposition (SVD). |
22.3 | |

22.4 The theory behind the singular value decomposition. Orthogonality and orthonormality. |
22.4 | |

22.5 Low rank approximations with the singular value decomposition. |
22.5 |