My question will probably be ill posed, I apologize in advance, I did not receive any formal education in college maths. I was wondering if there was a field of mathematics that studied how to reduce the information necessary to represent invariants, or to produce an optimal representation of structure of elements under certain constraints so that the structure is preserved under certain transformation.
For lack of a better way of conveying my meaning, is there a field of mathematics interested in the best way to represent this kind of "invariant" structure for example: Some structure you want to represent
Assuming the system has a grid like representation built in and you do not have to define it.
A non-optimal representation for example would be a list of x,y coordinates for each points. A better representation might be a N*N*2 distance matrix between each points in x and y ?
With such matrix form representation, invariance with respect to scaling would only be represented by a scalar multiplication.
Is there a formal way to 'look' for the 'minimum information' structure ?
But my interest goes beyond this example, this is a structure in space but what about a structure in space and time, what about if you add some characteristics to each points or build more complex relationships etc...
Is there a field studying such questions formally, and if yes what would be the best way to learn without going to school (a good course book you could recommend ?).
Thanks ! And I hope this made some sense...
A couple of points:
Before you can talk about representations of your object, you need to say what exactly your object is. Taking the example of your picture, is it relevant that all the vertices lie at lattice points? Do the angles and distances have meaning? If you turned it upside down would it be the same object or a different one? Is it fundamentally different from the '8' symbol?
Only once you've specified exactly what your object is can you talk about any of the other things you want to talk about.
Once you've identified precisely what your object is, you can pick some class of descriptions of it and then ask which is the "smallest" in some sense.
For instance, let's say that your object is a metric space. That is, it's a collection of five points with no particular order to them and the only thing relevant is the distances between the five points.
So let's do the following. Pick a (mathematically well-defined) computer programming language -- some particular way of encoding Turing machines might be a good choice. Consider the (infinite) set of computer programs that output a distance matrix corresponding to your metric space, in some specified format. Among the programs in this set are some of minimal length, and we can take this length as a measure of the complexity of your metric space.
This measure (made suitably formal) is called the Kolmogorov complexity. Of course the specific number depends on the particular format you fix both for the computer program and its output, but in practice you'd be more interested in asymptotic behavior than the value for some particular object.
There are many other notions along these lines. The general ingredients would be something like this:
For Kolmogorov complexity, these could be:
Another common measurement along these lines would have:
If we were to apply the analogous notion to vector spaces then our "complexity" would recover the notion of dimension, and our "minimal representation" would recover the notion of a basis.
These kinds of ideas are fairly ubiquitous in mathematics, and are in fact really synonymous with the idea of a minima in general, since we can think of any arbitrary set $S$ of objects as the set of representations of something. (For instance, we can consider a topological space $T = S \cup S'$ where the open sets are precisely $\emptyset, S, S', T$. We can take our notion of "representation" to be closure, and our notion of size to be cardinality. Then any element of $S$ is a minimal-size representative of $S$.)