[om-list] Dimensionality reduction

[Mark Butler]

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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