[om-list] Dimensions in knowledge space

[Mark Butler]

MrM0j0r15n Rajeev the kanehbosm wrote:

> I have a question regarding your discussion on dimensions in knowledge
> space. My questions is, wouldnt, any "non-normative" KRS based on dimensions
> of quality, immediately degenrate into an infinite dimensional knowledge
> space ? How does one utilise the useful concept of dimensions without going
> overboard ?

Yes, the number of dimensions is arbitrary.  Most dimensions (like words) are completely artificial, and once cannot pretend to represent everything there is to know about an object in terms of a finite number of attributes or dimensions.
 
> For, eg, if OM were to be used in say political analysis, we might arrive at
> a point where we need to "ask" OM, of how one event is similiar to another
> event. In my opinion (uneducated), we would now need an additional dimension
> just to formulate such a question. Going in this direction, where can we
> find the upper limit on the number of dimensions required ? 

There is no such limit, but no need to find it either.  One should strive to have as few as possible, but many dimensions are irreducible.  For example, friendship - friendship is a function of each pair of possible friends and there is no necessary correlation between friend(A,B) friend(B,C) and friend(A,C), which means that is least a three dimensional problem, if not a six dimensional problem.

The number of dimensions has no consequence - no fixed length vectors here. In any given calculation, we might simply identify the relevant shared dimensions and ignore all of the rest.

Shared dimensions imply a very specific relationship between any pair of objects.  More importantly, known relationships can be used to derive unknown relationships, which is the whole point.  Given the position vectors Rab and Rbc, Rac is easy to identify as the vector addition Rab + Rbc.  (*)

Even when dimensions are strictly independent, as in the case of the (N*(N+1))/2 relationships between N parties, there are very strong statistical correlations between similar relationships.  If A is a  likes B and B likes C, common experience surely indicatates that the probability that A will like C is greater than the probability that A will like someone chosen at random, notwithstanding extremely strong contrary experiences where A can't stand the sight of C.

Classical data modeling is all about determining a minimal set of non-redundant single-valued attributes to describe each member of a set of similar objects, which makes any data set from a well designed database well suited to this type of dimensional analysis.  The whole field of data mining is based on using techniques to inductively discover hidden relationships in large bodies of such data.  This is only practical because data in a typical database is highly normalized to begin with - if instead of having a nearly complete data set about a large number of objects, you had a random collection of uncorrelated propositions, one would have to normalize the data somehow to some reasonable set of analyzable dimensions to begin with.

I guess what I am saying is that the fact that objects share common dimensions is the only thing that makes them relatable or comparable in the first place, so viewing objects as points or extents in an arbitrarily dimensioned space, at least as an abstraction, seems to me to be a both a convenient perspective and one that appears to be complete enough to represent any real world problem.

- Mark

(*) It was Tom's idea to treat relationships between objects as multi dimensional vectors in a conceptual space, and more especially to apply geometric analysis to the relationships between those vectors, particularly in abstract dimensions where the existence of transitive geometric relationships between objects is not at all obvious.