Learning in commutative algebra & models for random algebraic structures

Abstract: A commutative algebraist's interest in randomness has many facets, of which this talk highlights two. Namely, we will discuss 1) how to use basic statistics and learning for improving Buchberger's algorithm and 2) how to generate samples of ideals in a `controlled' way. The two topics, based on joint work with various collaborators and students, form a two-step process in learning on algebraic structures, designed with the aim of avoiding the 'danger zone' of blind machine learning over uninteresting distributions. For learning, we show that a multiple linear regression model built from a set of easy-to-compute ideal generator statistics can predict the number of polynomial additions somewhat well, better than an uninformed model, and better than regression models built on some intuitive commutative algebra invariants that are more difficult to compute. We also train a simple recursive neural network that outperforms these linear models. Our work serves as a proof of concept, demonstrating that predicting the number of polynomial additions in Buchberger's algorithm is a feasible problem from the point of view of machine learning. As a first example of sampling, we present random monomial ideals, using which we prove theorems about the probability distributions, expectations and thresholds for events involving monomial ideals with given Hilbert function, Krull dimension, first graded Betti numbers, and present several experimentally-backed conjectures about regularity, projective dimension, strong genericity, and Cohen-Macaulayness of random monomial ideals. The models for monomial ideals can be used as a basis for generating other types of algebraic objects, and proving existence of desired properties.

machine learningalgebraic geometrynumber theory

Audience: researchers in the topic

DANGER: Data, Numbers, and Geometry