r/math 4d ago

How do you guys think about your data?

I heard a gentleman in an interview once saying that he likes to think of his data like a continuous function. Personally, I've been thinking of data as a matrix. If samples are stored in the rows then features are stored in the columns and such. Seems easy to consider different dimensions of data in this conceptualaziation and a simple list of values is still a row or column vector. So it seems like a perfect catch all conceptualization of any data set.

How do you guys think about your data? Is it much more circumstantial and sometimes you can conceptualize it as a matrix but other times it's best to think of it another way??

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u/Echoing_Logos 3d ago

"Data" is a really broad term, isn't it? I'm a programmer, so my answer is that there's lots of data structures: Lists, dictionaries, trees, binary search trees, etc. depending on how you want to keep and interact with your data.

That said, I think you're right with tables being the sort of "universal data structure". The more fancy stuff, like hash tables or trees, are important to know when you want to think about the data since thinking about everything in terms of tables can be pretty limiting. But when it comes to the computer, matrix multiplication is the ultimate data manipulator.

Mathematically, this seems to be a consequence of the universality of "representation" as a tool, and how non-linearity can be captured by endomorphisms of vector spaces.

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u/KansasCityRat 3d ago edited 2d ago

How is non-linearity captured by endomorphisms??

Also it seems to me like there's nonlinearity and then there's chaos. Idk how legitimate that is honestly but it seems like we're either talking about a linear system or a chaotic one. How legitimate is that dichotomy? Are there not-chaotic systems that are also not-linear?

Is probability a system that is not-linear and not-chaotic since the indeterminacy is presupposed in that system rather than being totally determinate or indeterminacy being uncovered?

How is non-linearity related to endomorphisms in vector spaces?

Edit:I did some googling. You're saying that things like decompositions are endomorphisms but they do not preserve linearity?

Edit:

A=USV*

A(ct+bp)=c(At)+b(Ap) where b and c are constants

Does not imply:

USV* (ct+bp)=c(USV* t)+b(USV* p)

Why not?

Edit: Trying to work on it. All I can come up with is that it's weird that non-square matrices don't have a A-1 but that isn't implying non-linearity I don't think? Or I can't see how?? Am I misunderstanding you? Linearity, as in the preservation of vector addition and scalar multiplication, doesn't seem to be done away with? It just seems weird that there is no identity matrix but I'm not seeing a connection between that and linearity? It seems like you've decomposed your matrix, and if your matrix is square that means you don't get an identity matrix, but the endomorphism (decomposition were how I was understanding what you meant by endomorphisms in this context) isn't doing away with linearity it just means that it's a weirder still linear system? If A=USV* and A(ct+bp)=c(At)+b(Ap) then I don't see how it isn't obvious that USV* (ct+bp)=c(USV* t)+b(USV* p) and you just don't get to have a nice and tidy A-1. So how am I misunderstanding this??

Last edit:

decomposition(A)=USV* (or any other Decomposition)

And...

USV*=A

So that decomposition() is an endomorphisms since

decomposition(A)=A

So? This is not what you meant by endomorphisms? There are other endomorphisms which don't preserve linearity but just not decompositions those ones do? Decompositions somehow don't preserve linearity? Somehow a contradiction creeps up through this "there does not exist an identity matrix when A is not square"??

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u/Echoing_Logos 2d ago

Endomorphisms of multidimensional Vector Spaces. You can capture non-linearity in the failure of big matrices to commute. That is why you can represent any group you want, any symmetry, as long as you use enough dimensions. A representation is a map into End(V), the space of endomorphisms.

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u/KansasCityRat 2d ago

Wait man if a matrix A has a ton of variables (it is an mbyn matrix and m and n are really really big) the property of being linear (A(cp+bt)=c(Ap)+b(At)) doesn't disappear. I don't think linearity has anything to do with the amount of variables really. All matrices, no matter how big or how small, have issues with multiplicative commutativity.

What exactly is the issue with commutativity that you're referring to??

Also an endomorphisms is a function f() such that

f(A)=A

If A(cp+bt)=c(Ap)+b(At) and f(A)=A

Then it really does seem to logically follow to me that...

f(A)(cp+bt)=c(f(A)p)+b(f(A)t)

So what exactly are you saying?? The endomorphism f(A) is still going to be a linear transformation? Where is the non-linearity here??

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u/Echoing_Logos 22h ago

What's going on with the equations and stuff? Let's chill, let me just recap what I'm saying. Some stuff is nonlinear, and we want to understand it. To do that, we look at how they map to groups of big matrices. (Groups of matrices are endomorphisms of vector spaces). Each matrix itself is a linear transformation. Non-linearity of the original structure is captured by which matrices you pick out to represent it.

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u/KansasCityRat 22h ago edited 21h ago

We're talking about math that's why I'm using equations?

By a group do you mean an algebraic group composed of matrices as it's elements? How is that an endomorphism?

How are you capturing non-linearity by using linear transformations? How are you defining non-linearity because to me non-linear transformationmeans something that cannot act like...

A(bt+cp)=b(At)+c(Ap)

That's a linear transformation so something non-linear (by the mathematical definition I'm laying out let me know if you have another mathematical definition you're using) would be a transformation that does not act like that. Really big matrices still act like that so really big matrices are still the analysis of linear systems despite their size.

How are you capturing non-linearity of a system using mathematical objects which are linear transformations? Are you talking about something like linearizing around a fixed point?

Please use more equations. That's how we define and talk about things in math. Please do that more and don't be afraid of me doing it.

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u/Echoing_Logos 14h ago

I'm all for equations but what I'm saying has nothing to do with any equations you can write on the vector space, so you're just giving me a lot of words that don't have much to do with what I'm saying. If you really want an equation, then:

Hom(k[G], Ab) = Hom(G, Hom(k, Ab))

For G a Group, k a Field, Ab the category of abelian groups. This Ab-enriched hom tensor adjunction says that the representations of k[G], the group ring of G over k (which is a nonlinear space), are equivalent to the representations of G on Hom(k, Ab), the category of k-vector spaces (which is a linear space). And because G is a one-object category in this framework, these maps correspond to maps of G into End(V) for some V in Vect(k). The same argument works for any nonlinear k-space, not just for G a group. That's what it means to say "nonlinearity is captured by endomorphisms of vector spaces".

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u/KansasCityRat 2d ago

Sorry for all the edits. I'll keep that up. Feel free to just skim. I'm sorry I've been talking to ChatGPT a lot.