I have heard about uses of Algebraic Geometry in Statistics and Machine Learning. I wanted to try to learn a bit about this topics. I don't know nearly anything about Algebraic Geometry, but I have background in math, and I know about basic group theory, rings fields and some commutative algebra. My questions are:

  1. What are the Algebriac Geometric concepts I should learn that are connected to applications in Stats/ML (I suppose only a portion of what is usually taught in Algebraic Geometric courses and books is useful).

  2. Can you recommend some books / introductory papers for someone with my background? I don't mean standard textbooks for AG but something that focuses on concepts used in applications.


2 Answers 2


Here is a list of the standard references:

Here is a list of related references, not directly addressing algebraic statistics, although providing background in the methodology used for the subject:

Web pages of courses about the topic, past and present:

These lists are almost certainly by no means comprehensive.


$\dagger$ links dead and not archived

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    $\begingroup$ When you visit again here, could you provide alternate links to those that are dead? I have been able to amend some but not all. $\endgroup$ Commented Apr 6 at 4:36
  • $\begingroup$ @User1865345 I wasn't able to find links for those either. I found Professor Shane Kelly's current website, but the seminar wasn't listed in his past teaching, I guess because it was too far back in his career or it was too informal. I couldn't find a link for the TU Berlin course either. The webpage seems to have belonged to Fatemeh Mohammadi (slmath.org/people/22091) but the course isn't listed on her current website's teaching page fatemehmohammadi.com/teaching (although similar courses are). $\endgroup$ Commented Apr 13 at 13:43
  • $\begingroup$ Thanks for the update, Chill2Macht. $\endgroup$ Commented Apr 13 at 13:44

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Sure to be influential, this book lays the foundations for the use of algebraic geometry in statistical learning theory. Many widely used statistical models and learning machines applied to information science have a parameter space that is singular: mixture models, neural networks, HMMs, Bayesian networks, and stochastic context-free grammars are major examples. Algebraic geometry and singularity theory provide the necessary tools for studying such non-smooth models. Four main formulas are established:

  1. the log likelihood function can be given a common standard form using resolution of singularities, even applied to more complex models;

  2. the asymptotic behaviour of the marginal likelihood or 'the evidence' is derived based on zeta function theory;

  3. new methods are derived to estimate the generalization errors in Bayes and Gibbs estimations from training errors;

  4. the generalization errors of maximum likelihood and a posteriori methods are clarified by empirical process theory on algebraic varieties.

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    $\begingroup$ Thank you for the reference -- this book looks like the best recommendation so far! I will have to get it soon. Also for anyone interested here is a question on this website about this book: stats.stackexchange.com/questions/22391/… $\endgroup$ Commented Oct 20, 2016 at 16:59

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