[om-list] Dimensionality reduction

Wed Apr 16 16:23:18 EDT 2003

Interesting little page.

http://www.cs.mcgill.ca/~sqrt/dimr/dimreduction.html

tomp

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> Tom,
> 
>   Linear Algebra / Vector Analysis covers the basics of what you are 
> talking about.  Engineering majors typically take it in their junior 
> year. QM uses a infinite "dimensional" generalization of vector algebra 
> called Hilbert spaces, but that level of complexity is not necessary 
> unless you are analyzing continuous rather than discrete functions. (The 
> idea is to treat a function just like a vector, except with continuously 
> valued instead of discretely valued indexes).  String theory is *much* 
> further off the beaten path, and still lies in the realm of informed 
> speculation, from what I can tell.
> 
> In any case, linear vector analysis covers the mechanics of transforming 
> from one set of basis vectors to another, which is generally 
> accomplished by multiplying with a linear transformation matrix. 
>  Specific reduction algorithms are not discussed in such classes (or 
> textbooks), but they will give you a good basis for understanding books 
> and papers written in the field of dimensionality reduction, such as 
> this one:
> 
>    Geometric Data Analysis: An Empirical Approach to Dimensionality 
> Reduction and the Study of Patterns,
>    Michael Kirby, Wiley, 2000
> 
>    http://www.wiley.com/cda/product/0,,0471239291,00.html
> 
>  - Mark B.
> 
> Thomas L. Packer at home wrote:
> 
> >     Also, could you tell me if there is a name for a branch of 
> > mathematics that deals with many dimensional spaces, especially a 
> > formalism that has the ability to take information about points or 
> > phenomena in space and to calculate the minimum number of dimensions 
> > necessary to represent that information?  I've heard of such a 
> > challenge accomplished in quantum mechanics and/or string theory; but 
> > I'm not sure how formally or mathematically it was done.  That is, 
> > I've heard that there are like 7 dimensions necessary to explain ... 
> > is it string theory or QM?  I can't remember.
> 
> 
> 
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> <html>
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> Tom,<br>
> <br>
>   Linear Algebra / Vector Analysis covers the basics of what you are talking
> about.  Engineering majors typically take it in their junior year. QM uses
> a infinite "dimensional" generalization of vector algebra called Hilbert
> spaces, but that level of complexity is not necessary unless you are analyzing
> continuous rather than discrete functions. (The idea is to treat a function
> just like a vector, except with continuously valued instead of discretely
> valued indexes).  String theory is *much* further off the beaten path, and
> still lies in the realm of informed speculation, from what I can tell.<br>
> <br>
> In any case, linear vector analysis covers the mechanics of transforming
> from one set of basis vectors to another, which is generally accomplished
> by multiplying with a linear transformation matrix.  Specific reduction algorithms
> are not discussed in such classes (or textbooks), but they will give you
> a good basis for understanding books and papers written in the field of dimensionality
> reduction, such as this one:<br>
> <br>
>    Geometric Data Analysis: An Empirical Approach to Dimensionality Reduction
> and the Study of Patterns, <br>
>    Michael Kirby, Wiley, 2000<br>
> <br>
>    <a class="moz-txt-link-freetext" href="http://www.wiley.com/cda/product/0,,0471239291,00.html">http://www.wiley.com/cda/product/0,,0471239291,00.html</a><br>
> <br>
>  - Mark B.<br>
> <br>
> Thomas L. Packer at home wrote:<br>
> <blockquote type="cite" cite="mid005f01c303ae$ec1c5b20$ef60070a at TOMP4">  
>   <meta http-equiv="Content-Type" content="text/html; ">
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>   <div><font color="#008000" size="2"></font></div>
>  
>   <div><font color="#008000" size="2">    Also, could you tell me if  there
> is a name for a branch of mathematics that deals with many dimensional  spaces,
> especially a formalism that has the ability to take information about  points
> or phenomena in space and to calculate the minimum number of dimensions  necessary
> to represent that information?  I've heard of such a challenge  accomplished
> in quantum mechanics and/or string theory; but I'm not sure how  formally
> or mathematically it was done.  That is, I've heard that there are  like
> 7 dimensions necessary to explain ... is it string theory or QM?  I  can't
> remember.</font></div>
>  
>   <div></div>
> </blockquote>
> <br>
> </body>
> </html>
> 
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