[om-list] Dimensionality reduction

[Thomas L Packer]

Mark

   One long-winded question to make sure we are all on the same page.  Is the
thing were are talking about one method, or two similar methods?

   What I see in the web sources I've found on dimensionality reduction, they
look simply like machine learning: a synthetic (inductive) method of trying to
find patterns in many data points.  What I'm most interested in finding right
now is a purely analytic (deductive) method of taking a set of data-points
measured in space, probably using more dimesions than are necessary, and using
this method to analytically reducing that data into an equivalent picture using
as few dimensions as possible.  The key here is: no information loss.  Most or
all machine learning is a fuzzification of data, which information lost.

  I'm thinking that there must be some technique for doing this in order for
scientists to be so confident about the exact number of dimensions needed to
account for certain physical phenomena.

   For example of something related to what I'm talking about: say you have look
at the stars at night, and take readings of planet's positions through time.  So
you have a bunch of data in three dimensions: two spatial and one temporal
dimension.  You would like to reduce (or maybe translate or rotate) those data
into three purely spatial dimensions.  I know some astronomer has done this kind
of thing, and I'm guessing that it was purely analytic, not synthetic.

  As a better example, say you are investigating the behaviour of flying
saucers, and you see a flock of them hovering in the sky.  You have two
competing hypotheses: (1) they orient themselves in a straight line, (2) they
orient themselves in a plane.  But your own movement is restricted to certain
places on the ground.  So you make the best use out of your movement as you can
by walking around and taking measurements of the saucer's locations in your
field of vision.  After you are done you have, you have a pile of data.  Is
there an analytic mathematical formula that will tell you (1) whether you may
conclude that the flying saucers always hover together in a linear pattern, and
(2) what the location of that linear pattern was, and the location (in
three-dimensions) of each saucer?

  Assuming ideal conditions (a perfectly straight line or plane), there should
be no information loss.

   It's totally possible that this is the same thing as what you have been
talking about.  I just wanted to make sure.

  Thanks for the feedback.

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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> <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
> <html>
> <head>
>   <title></title>
> </head>
> <body>
> 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; ">
>  
>   <meta content="MSHTML 6.00.2800.1141" name="GENERATOR">
>   <DEFANGED_STYLE_0 <=""> 
>   <style></style>  </DEFANGED_STYLE_0>
>   <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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> 
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