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Matrices and Graph

thepalindrome.org

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Re: Matrices and Graph

#21

Post by carlossouza »

EsportToys wrote: post #9

Fun fact: this is only valid for domains that have a notion of "selfness", i.e. that there is such thing as an "identity matrix" for the quantities. Consider the following square matrix: TSLA APPL GOOG MSFT Alice | 100 5 0 1 Bob | 0 30 100 5 Carol | 2 2 2 2 Dan | 0 0 0 1000 An input vector of stock prices gives an output vector of net worths. However, that is about the only way you can use this matrix. You cannot tra…

I disagree... there are more ways you can use this matrix to creatively extract information out of it.

For instance, you can normalize along the columns, and build a "recommender system" using matrix factorization.

With that, when a new person comes with a portfolio, the system will output a probability for this new person to acquire the other assets he doesn't have.

It's (the very basic) idea of how Netflix recommends movies.

Re: Matrices and Graph

#23

Post by [deleted] »

derbOac wrote: post #20
EsportToys wrote: post #9

Fun fact: this is only valid for domains that have a notion of "selfness", i.e. that there is such thing as an "identity matrix" for the quantities. Consider the following square matrix: TSLA APPL GOOG MSFT Alice | 100 5 0 1 Bob | 0 30 100 5 Carol | 2 2 2 2 Dan | 0 0 0 1000 An input vector of stock prices gives an output vector of net worths. However, that is about the only way you can use this matrix. You cannot tra…

You make a good point about types of matrices that a graph representation makes sense with but it seems a bit much to say that PCA is pseudoscience? If you had a lot of people and a lot of stocks, a low-rank representation of the matrix (probably not PCA per se with that particular matrix, but something closely related) could convey a lot of information about, e.g., submarkets and how they're valuated together. Or no…

[deleted]

Re: Matrices and Graph

#24

Post by an_d_rew »

EsportToys wrote: post #9

Fun fact: this is only valid for domains that have a notion of "selfness", i.e. that there is such thing as an "identity matrix" for the quantities. Consider the following square matrix: TSLA APPL GOOG MSFT Alice | 100 5 0 1 Bob | 0 30 100 5 Carol | 2 2 2 2 Dan | 0 0 0 1000 An input vector of stock prices gives an output vector of net worths. However, that is about the only way you can use this matrix. You cannot tra…

When I try to get this point across about techniques like the PCA, I like to show that the measurement units strongly affect the inference.

Really, if your conclusions change depending on whether you measure in inches or centimeters, there’s something wrong with the analysis!

Re: Matrices and Graph

#26

Post by enchiridion »

I think I’m misunderstanding. The node relabeling seems backwards.

He says start with the highest order, which makes me think the neighborhood with order 3 would get the smaller node labels, and the neighborhoods with order 0 would get the highest.

It looks like the opposite was done.

Re: Matrices and Graph

#28

Post by meindnoch »

Scene_Cast2 wrote: post #15

Another interesting mapping is that a vector is (or can be thought of as) a discrete function (f(x) = ....) over an interval, a dot product of two vectors is a discrete integral product, and a matrix is a discrete scalar field. I wonder what the continuous form of a graph is... Some sort of a manifold perhaps?

Extremal graph theory models graphs with an infinite number of vertices as real-valued functions on the unit square. Book: https://lovasz.web.elte.hu/bookxx/hombook-almost.final.pdf

Re: Matrices and Graph

#29

Post by owlbite »

zmgsabst wrote: post #7

This is especially fascinating when you consider graphs/diagrams are a way to encode math.

The fact that you can represent a graph (the mathematical abstract object) as a diagram is sort of by-the-by here.

The most important thing is that graph algorithms and concepts have a strong relation to numerical aspects of the linear algebra and can be used to accelerate computation.

(You could of course argue that the act that graphs can be represented as a diagram helps humans come up with such algorithms, but that's basically equivalent to saying that you can represent a matrix as a block of numbers and that helps humans look at it).

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