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-   -   *ANSWER* Q10 maybe (http://book.caltech.edu/bookforum/showthread.php?t=4235)

heeler 04-23-2013 09:54 AM

*ANSWER* Q10 maybe
 
On question 10 I found it hard to visualise how the pattern changed until I realised I could conformally map the problem given into one of the examples. This left me wondering if this is generally true for learning models.

I used the r = \sqrt{x_1^2 + x_2^2} to transform the problem from \mathbb{R}^2 to \mathbb{R}.

It seems like the connection is there but I wanted to ask in case I was leading myself astray.

yaser 04-23-2013 09:59 AM

Re: *ANSWER* Q10 maybe
 
Quote:

Originally Posted by heeler (Post 10561)
On question 10 I found it hard to visualise how the pattern changed until I realised I could conformally map the problem given into one of the examples. This left me wondering if this is generally true for learning models.

I used the r = \sqrt{x_1^2 + x_2^2} to transform the problem from \mathbb{R}^2 to \mathbb{R}.

It seems like the connection is there but I wanted to ask in case I was leading myself astray.

This is true here because the circles are concentric. In general, it may not be possible to reduce the learning model to an equivalent one-dimensional version.

Elroch 04-23-2013 10:09 AM

Re: *ANSWER* Q10 maybe
 
Quote:

Originally Posted by yaser (Post 10564)
This is true here because the circles are concentric. In general, it may not be possible to reduce the learning model to an equivalent one-dimensional version.

There's something about this problem that seems to make a lot of us think "can this really be right?" :) I think we agree it is. One way I thought of it was to observe that the hypothesis set can never separate any two points at the same radius. This means you only need to consider one representative point at each radius, without loss of generality.

This has an analogy to an idea in topology that when points share all the same neighbourhoods they are effectively like the same point, and a quotient space can be formed which merges all the unseparable points with each other. I feel there may be the potential for more connections between hypothesis sets and topology to be drawn, although there are major differences as well as similarities.


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