These days, I am reading and learning about Category Theory. To be honest it is yet very abstract for me, specially those parts which go to Topos Theory, Algebraic Geometry and Algebraic Topology. But I have this feeling that this is the direction I would really like to take in future.
In terms of modeling, I think they will go to cases where each point (let say each observation) is not a real valued point, but an abstraction of a point. Let say, instead of having one point in cartesian space (regardless of the dimensionality) we might have for example a set of distinct points to just represent one observation. I am speculating now, but I think this should go to Quantum Observations, I guess.
What I am currently focused on in my PhD thesis is the interpretation of Universals from Set Theoretical view which goes to Abstract Universals and Category Theoretical view which corresponds to Concrete Universals. Relating these issues to the problem of modeling, following is my current take…
PCA assumes that the world is constructed from orthogonal dimensions, just like a set of consistent rule set. No overlap and no dependencies. So, world is very rational and always we have ideal forms, but individuals are not following the rules, they are deviating from the ideals. Therefore, they are Erroneous… and this is when the concept of Error was born. Therefore, just imagine you have a set of data points. Fit a PCA based model. You get something like this:
On the other hand SOM says lets take an opposite view, ideals are what ever the individual agreed on. So there is no a priori ideal form. Universals are coming out of concrete individuals, opposite to PCA where Universals are Abstract. So, in the above data set if you want to explain the state space with SOM we might get something like this:
But again SOM says, if you give me lots of individual (concrete) cases, I make their own forms, with no external references. So, if you give SOM thousands of curves, it gives you the Eigen curves as follows.
And it is not just a visualization, it is a computable symbolic space.