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There are some books here, but all focus on linear models.

Do we have books/notes with broader coverages?

In case of linear models, “coordinate-free” basically means “matrix-free” and uses the theory of vector space, and the viewpoint is geometric. (Outside of linearity, would we go into something like Information Geometry?)

Edit: It seems Information Geometry and Topological Data Analysis are related to this question. But books on IG and TDA tend to collect recent research results, and the problems solved are specialized. Rather, I’d like to see coordinate-free reconstructions of main stream statistical methods (maybe also advanced ones), and this would also help to learn IG and TDA, proper.

wpzdm
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    You might want to look up Topological Data Analysis in addition to Information Geometry. – Tristan Duquesne Sep 22 '21 at 10:48
  • @TristanDuquesne Thank you! While both topics are intersting, I am asking this question mainly for learning purpose. Most IG and TDA books seem to be very advanced. – wpzdm Sep 22 '21 at 10:59
  • What/why are you looking to learn specifically ? Most coordinate-free approaches in mathematics require an abstract understanding of the objects being manipulated, and this generally requires to consider "pure math" formulations. Eg, I've seen textbooks of linear algebra here and there that rely on category theory to better prepare subjects like differential geometry or geometric calculus (most notably to broach the notion of a quotient space, which requires a coordinate-free approach, and with it, rigorously construct the major quotients of the tensor algebra over $V$). – Tristan Duquesne Sep 22 '21 at 11:14
  • @TristanDuquesne Thank you for the following up! I think the thing is the expected coverage of topics of the books. I updated the question. I am willing to learn “pure math.” I learned some abstract linear algebra (see https://math.stackexchange.com/questions/4246665/) and want more. And I am also learning category theory! – wpzdm Sep 22 '21 at 11:23
  • Note: I found the article "Persistent Homology: An Introduction and a New Text Representation for Natural Language Processing" by Xiaojin Zhu, and "Introduction to Topological Data Analysis" by Julien Tierny to be very "short and sweet" introductions to TDA for someone with a decent understanding of calculus and abstract algebra. If you're lacking in your understanding of either, feel free to contact me via the address on my github account (in order not to turn the comments into a full conversation): https://github.com/Fulguritude/ for more reading/resource suggestions – Tristan Duquesne Sep 22 '21 at 11:42

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