Advanced Linear Algebra - Lecture 41: Low Rank Approximation and Image Compression

We introduce the Eckart-Young-Mirsky theorem, which says that the singular value decomposition (or, equivalently, the orthogonal rank-one sum decomposition) can be used to find the closest low-rank matrix to a given matrix. We then show that this theorem can be used to (lossily) compress an image.

Please leave a comment below if you have any questions, comments, or corrections.

Timestamps:
00:00 - Introduction
03:12 - Eckart-Young-Mirsky theorem
05:46 - 3x3 example
09:49 - Image compression

[dzanc]

r/math led me here and this is great. This topic was briefly touched during a quantum information course I attended last year when dealing with renormalization group techniques and indeed the core idea is the same, keep the largest populations/eigenvalues/what have you to extract the most information with a given amount of resources

[JohannesSchmid-hmic]

That was the best linear algebra lecture series I ever watched and the best I will ever watch! Thank you!

[michaelmuller]

Went through your introductory and now advanced LA playlist, very informative, thank you!

[supersnowva]

Another excellent lecture!!! The last minute of illustration on the color banding effect blew my mind!!! Thank you so so much!

[quantabot]

Im here, Going to listen again. Best linear algebra teaching i have seen so far

[weneedlittlepatience]

I've been watching your videos since the first one of introductory L.A. all the way to this one, and thanks to you I feel really excited about how this is related to AI/ML/DL

[caseyj]

Thank you! You really helped clarify my Matrix Theory course!

[vijayarana]

Thanks for the amazing explanation !!!! Now the subject feels interesting

[jamesyang]

Hello, Nathaniel, wonder explanation for the LoRA application in image compression. For R=1, it is easy to understand that U1 is the x-axis and V1 is the y-axis which are orthogaonal to each other. I am wonder how R=2 (only), R=3 (only) look like ? which U2 (U3) and V2 (V3) must represent certain diagonal patterns in the 2-D image. Thank you👍

[DrAndyShick]

Hi, I know this was a while ago, but do you plan on doing any lectures on multilinear/tensors?

[FuzzyBagels]

THANK YOU!!! I was able to finsh my final project after watching this lol