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Stream: theory: category theory

Topic: Matrix trace as feedback

Adam Nemecek (Aug 14 2023 at 01:00):

It is said that that the trace in the context of traced monoidal categories is generalization of a matrix trace. I'm having a hard time seeing how the matrix trace is a feedback loop though. I came across this paper
grab7449.png but the formula seems wrong.

JS PL (he/him) (Aug 14 2023 at 03:56):

Adam Nemecek said:

It is said that that the trace in the context of traced monoidal categories is generalization of a matrix trace.

This is true when you consider the trace operator with respect to the monoidal product $\otimes$ of the categories of matrices and not the biproduct $\oplus$. In the sense that the category of matrices over a ring is a traced monoidal category and some of the axioms of the trace operator, such as dinaturality, correspond to basic properties of the matric trace, like the cyclic property.

JS PL (he/him) (Aug 14 2023 at 03:58):

Here's a screenshot of some notes I wrote a while ago, which explains how the classical trace operation for matrices gives a trace operator on the tensor product $\otimes$ of matrices.
Firefox_Screenshot_2023-08-14T03-57-11.115Z.png
You get back the classical trace of an $n \times n$ square matrix $A$ by considering it as an endomorphism $A: n \to n$ and then taking the trace of the unit $I= 1$, giving you a map $Tr(A): 1 \to 1$ which is just a scalar, so $Tr(A) \in R$

JS PL (he/him) (Aug 14 2023 at 04:02):

Adam Nemecek said:

I came across this paper
grab7449.png but the formula seems wrong.

In this paper, they are talking about a partial trace operator on the biproduct $\oplus$ for the category of matrices. It's a partial trace since this formula only holds when $Id - A$ is invertible of when the infinite sum $\sum A^n$ converges

JS PL (he/him) (Aug 14 2023 at 04:03):

In general, if you have a category with biproducts and infinite sums, you have a general formula for a trace operator given by $Tr( \begin{bmatrix} A & B \\ C & D \end{bmatrix} ) = A + \sum B D^n C$

Adam Nemecek (Aug 14 2023 at 04:21):

thanks for the answer. do you have any good resources on this?

JS PL (he/him) (Aug 14 2023 at 04:25):

Adam Nemecek said:

thanks for the answer. do you have any good resources on this?

Happy to help. And certainly. Though resources on what part exactly?

Adam Nemecek (Aug 14 2023 at 04:25):

i think id like to know more about the connection between this general trace and finvects

JS PL (he/him) (Aug 14 2023 at 04:27):

Just to clarify which trace you mean: the one on the tenor product $\otimes$ or the biproduct $\oplus$?

Adam Nemecek (Aug 14 2023 at 04:35):

i guess the former

JS PL (he/him) (Aug 14 2023 at 04:36):

Alright well a good place to start is with this survey paper by Peter Selinger: https://www.mscs.dal.ca/~selinger/papers/graphical.pdf

Adam Nemecek (Aug 14 2023 at 04:36):

ok thanks

Adam Nemecek (Aug 14 2023 at 04:37):

this is the feedback trace right?

Adam Nemecek (Aug 14 2023 at 04:37):

if the tensor product is the feedback trace, what is the biproduct trace then?

JS PL (he/him) (Aug 14 2023 at 04:39):

Then here is the background section on traces I wrote for a paper:
tracenotes.pdf

JS PL (he/him) (Aug 14 2023 at 04:39):

Adam Nemecek said:

if the tensor product is the feedback trace, what is the biproduct trace then?

In my mind, I would say that the feedback trace is more the trace on (bi/co)products:
trace for products = feedback via fix points
trace for coproducts = feedback via iteration
trace for biproducts = feedback via iteration/fix points

Adam Nemecek (Aug 14 2023 at 04:41):

that actually makes a lot of sense

Adam Nemecek (Aug 14 2023 at 04:41):

can you think of a way this can be translated into programming?

Adam Nemecek (Aug 14 2023 at 04:41):

i had this idea that i need to think about some more

Adam Nemecek (Aug 14 2023 at 04:41):

but it feels as if this was roughly like iteration over a double ended queue

Adam Nemecek (Aug 14 2023 at 04:42):

a double ended queue is a circular buffer of sorts

Adam Nemecek (Aug 14 2023 at 04:42):

and like you can be inserting things at both ends as you are iterating over the elements

Adam Nemecek (Aug 14 2023 at 04:42):

i need to think some more about what this really means and what other operations are needed

JS PL (he/him) (Aug 14 2023 at 04:42):

Adam Nemecek said:

if the tensor product is the feedback trace, what is the biproduct trace then?

That said, compact closed also allow for feedbacks in a way. Graphically speaking at least

Adam Nemecek (Aug 14 2023 at 04:42):

and most imporatntly what is the advantage here

JS PL (he/him) (Aug 14 2023 at 04:43):

Adam Nemecek said:

can you think of a way this can be translated into programming?

No unfortunately programming is outside of my knowledge.

JS PL (he/him) (Aug 14 2023 at 04:44):

I suggest looking at Masahito Hasegawa's papers on trace monoidal categories.
Possibly this one as well: https://www.sciencedirect.com/science/article/pii/S0022404999001802 (though this one is in a much more general setting -- so possibly not very useful for what you want to do )

JS PL (he/him) (Aug 14 2023 at 04:45):

Maybe this paper might also help: http://www.numdam.org/item/ITA_2002__36_2_181_0/

Adam Nemecek (Aug 14 2023 at 04:49):

have come across some of these but didnt' quite take the plunge yet. will do

Adam Nemecek (Aug 14 2023 at 04:49):

thanks