Lecture 17: Rapidly Decreasing Singular Values

MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, Spring 2018
Instructor: Alex Townsend

Professor Alex Townsend gives this guest lecture answering the question 'Why are there so many low rank matrices that appear in computational math?' Working effectively with low rank matrices is critical in image compression applications.

License: Creative Commons BY-NC-SA

[philoneill]

This is a heck of a lecture. The flags example is really great. Thanks Professor Townsend.

[amirmotmaen]

Would love to see more from Alex Townsend. What a fantastic lecturer!

[darkwingduck]

WOw, Unbelievable and incredible handwriting! So clear, concise, and elegant!!

[georgesadler]

professor Alex Townsend, thank so much for a very theoretical background of numerical linear algebra at MIT. DR. Gilbert Strang has created a powerful course in linear algebra.

[aungkyaw]

The closing statement is to die for 😁 thanks for the amazing lecture.

[dpSimi]

Whelp. He's right that he's staying home and not driving around this year.

[learningstatistics]

16:45 why the second figure (square) 's rank is bounded by 1? I think that should be bounded by 2.

[ajitsamudrala]

3:38 Did he miss multiplication by singular values?

[raphaelambrosiuscosteau]

The key part of Zolotarev numbers explanation (sets of eigs are separate), unfortunately dosen't explain anything for me. Can anybody give a good intuition, why Zolotarev boundary works? The smoothnes at least makes sense, but while i kinda feel what Sylvester eq does, the end result is kinda deep math for me.

[BorrWick]

Where can we find 18.06 from Alex Townsend? I would love to see more lectures from him, such a rich explanation

[angfeng]

MIT has better blackboards than 20 years before when 18.06 was filmed.

[rogiervdw]

@3:30, why is he omitting the sigma scalars (singular values) in this sum of left & right singular vectors to produce X? Is that right?

[thaddeuspawlicki]

He has excellent blackboard penmanship.

[tmusic]

@ t = 8:45 If you transform the full rank Scottish flag i.e. turn it 45 degrees you get the rank 2 English flag. In other words, you can send the English flag with the instruction to turn it 45 degrees to avoid a full rank transmission.

[HanHan-hjpz]

why is a numerical low rank matrix hard to invert? he mentioned it briefly when introducing the vandermonde matrix but didn't clarify later.

[GauravSharma-uiyd]

Hope you are alright professor alex. You are really a star in this field.
I will probably work on this number once I reach 32.

[Enerdzizer]

16:15 why interior of Japan flag (second matrix) has rank 1? Maybe it's 2?

[justpaulo]

Prof. Alex: The world is Sylvester.
Tweety: Hey! How about me?

[MO-xikv]

Wow he call’s Prof. Strang “Gil”... I wish I was on these kinda terms with Gilbert.

[nickrichardson]

I miss Strang, but this is one hell of a lecture!