Help understanding this video about vectors in Machine Learning

[Nemanjavuk69]

I am currently following a long this video, which is an introduction to Machine Learning. (

)

At 47:15 to 48:45 the professor goes on to talk about, how "non normalizing" (sorry, I have a hard time hearing him even with subtitles on its a mess) the dimension of the vector goes down by 1. So having a vector in the realm of [imath]R^p[/imath] gives a dimension of [imath]R^{p-1}[/imath]. He even uses the example of if a vector was in [imath]R^2[/imath] than it would be [imath]R^1[/imath]. This seems wrong, but it might be me who have a hard time understanding him. Can someone brighter than me shine some lights on this? Thank you very much.

 

[Nemanjavuk69]

For some reason the site says the video is not avaiable. Here is the link again
https://-----www.youtube.com/-----watch?v=esTIhqAFKu4&list=PLGd9Gn0_Oc65os52iy4jwvex2q50r5q0U&index=27&t=2818s

Delete all the "-" I have inserted (there is only 10 of them) to have the actual link you can post in your URL. I have no idea why the site will not accept my original URL?

 

[nasi112]

Even if I wear a telescope glasses, I cannot see what is written on the board!

Write a question on vectors in Machine Learning that you could not answer or you had doubts of.

 

[Nemanjavuk69]

Even if I wear a telescope glasses, I cannot see what is written on the board!

Write a question on vectors in Machine Learning that you could not answer or you had doubts of.

I seem to be able to read it. I you perhaps watching the video on a phone? Furthermore, it is not so much the written thing but what the professor is saying. I seem to hear "non normalized" but I am suspecting him of trying to say "norm normalized", could this perhaps be it?

 

[The Highlander]

For some reason the site says the video is not avaiable. Here is the link again
https://-----www.youtube.com/-----watch?v=esTIhqAFKu4&list=PLGd9Gn0_Oc65os52iy4jwvex2q50r5q0U&index=27&t=2818s

Delete all the "-" I have inserted (there is only 10 of them) to have the actual link you can post in your URL. I have no idea why the site will not accept my original URL?

Just add your (YouTube?) link to (highlighted) text in your post (using the chain links icon above)

For example: Your Introduction to Machine Learning video.

The "Media" link has problems; avoid it. ?

Hope that helps. ?

 

[Steven G]

The writing IS small.

 

[nasi112]

I seem to be able to read it. I you perhaps watching the video on a phone? Furthermore, it is not so much the written thing but what the professor is saying. I seem to hear "non normalized" but I am suspecting him of trying to say "norm normalized", could this perhaps be it?

At what time, did he say that?

 

[Nemanjavuk69]

At what time, did he say that?

I already posted the timestamp at where the professor goes on and talks about this stuff which is 47:15 to 48:45, however, the first time he uses the sentence is 47:22 to 47:25, and then keeps using it throughout the the remaining of the video.

Furthermore, if you were to see the video on a smarthphone or tablet, YouTube allows you to zoom in on videos hence making the text much more clearer, on PC, you need to download an extension for it. Hope this helps.

 

[khansaheb]

The writing IS small.

That's because s/he did not want it to be on the wall...???

 

[nasi112]

I already posted the timestamp at where the professor goes on and talks about this stuff which is 47:15 to 48:45, however, the first time he uses the sentence is 47:22 to 47:25, and then keeps using it throughout the the remaining of the video.

Furthermore, if you were to see the video on a smarthphone or tablet, YouTube allows you to zoom in on videos hence making the text much more clearer, on PC, you need to download an extension for it. Hope this helps.

Can you add the video link here as suggested by post #5, so we can access it directly?

 

[blamocur]

I am currently following a long this video, which is an introduction to Machine Learning. (

)

At 47:15 to 48:45 the professor goes on to talk about, how "non normalizing" (sorry, I have a hard time hearing him even with subtitles on its a mess) the dimension of the vector goes down by 1. So having a vector in the realm of [imath]R^p[/imath] gives a dimension of [imath]R^{p-1}[/imath]. He even uses the example of if a vector was in [imath]R^2[/imath] than it would be [imath]R^1[/imath]. This seems wrong, but it might be me who have a hard time understanding him. Can someone brighter than me shine some lights on this? Thank you very much.

All he says that if you have a "non-normalized" [imath]p[/imath]-dimension vector [imath]x \in \mathbb R^p[/imath] then its normalized version [imath]\hat x = \frac{x}{||x||}[/imath] belongs to a [imath]p-1[/imath]-dimensional sphere [imath]\mathbb S^{p-1}[/imath], not [imath]\mathbb R^{p-1}[/imath]. Does this make sense?

 

[Nemanjavuk69]

All he says that if you have a "non-normalized" [imath]p[/imath]-dimension vector [imath]x \in \mathbb R^p[/imath] then its normalized version [imath]\hat x = \frac{x}{||x||}[/imath] belongs to a [imath]p-1[/imath]-dimensional sphere [imath]\mathbb S^{p-1}[/imath], not [imath]\mathbb R^{p-1}[/imath]. Does this make sense?

Could you perhaps ellaborate with the sphere part? Moreover, is it possible if you could illustrate it for me? It would greatly help. Furthemore, what is the difference between “norm normalized” and “non normalized”, it seems they both follow the same properties for the standard Euclidean length/norm function written ||⋅||. Quote me if I am wrong.

 

[blamocur]

Could you perhaps ellaborate with the sphere part?

What kind of elaboration do you need? Do you agree that [imath]\frac{x}{||x||}[/imath] has a norm of 1 for every non-zero vector [imath]x[/imath]? Or that all vectors with norm 1 lie on [imath]\mathbb{S}^{\,p-1}[/imath]. Please be more specific in your questions

Furthemore, what is the difference between “norm normalized” and “non normalized”,

I am guessing the difference is purely acoustical I.e., I believe he is saying "non-normalized".

 

[Nemanjavuk69]

Or that all vectors with norm 1 lie on S p−1\mathbb{S}^{\,p-1}Sp−1.

This one I want elaborated please. I can not see this for me. Could you perhaps also visualize it? What if the vectors were 2, 3 snd 4 dimensional, how would they look on the sphere? Again, visualizing this for me would be superb for my learning. Thanks in advance

 

[blamocur]

This one I want elaborated please. I can not see this for me. Could you perhaps also visualize it? What if the vectors were 2, 3 snd 4 dimensional, how would they look on the sphere? Again, visualizing this for me would be superb for my learning. Thanks in advance

Which part do you have problems with? That [imath]\frac{x}{||x||}[/imath] has the norm of 1? Here is why: for any scalar [imath]a[/imath] we have (by the definition of a norm): [imath]\left\| a x\right\| = a\left\|x\right\|[/imath]. Now use [imath]a = \frac{1}{\|x\|}[/imath]

As for vectors with norm 1 in [imath]\mathbb R^p[/imath] lying on [imath]\mathbb S^{p-1}[/imath] this is a definition of a sphere. I.e., [imath]\mathbb S^{p-1}[/imath] is a set of points at distance 1 from the origin. The distance from the origin is the same as the norm of the corresponding vector.

Not sure why you need visualization, but here is one for [imath]p=2[/imath] :