Computational complexity of learning algebraic varieties

TL;DR

This paper investigates the algebraic complexity of fitting algebraic varieties to point configurations, providing explicit formulas for spheres in one dimension and conjectures for higher dimensions, supported by numerical evidence.

Contribution

It introduces the algebraic complexity measure based on EDdegree for variety fitting problems and derives explicit formulas for spheres in one dimension, proposing conjectures for higher dimensions.

Findings

Closed-form formula for 1D sphere fitting complexity

Numerical experiments supporting conjectured formulas for higher dimensions

Analysis of algebraic complexity growth with number of points

Abstract

We analyze the complexity of fitting a variety, coming from a class of varieties, to a configuration of points in $\Bbb C^n$. The complexity measure, called the algebraic complexity, computes the Euclidean Distance Degree (EDdegree) of a certain variety called the hypothesis variety as the number of points in the configuration increases. For the problem of fitting an $(n-1)$-sphere to a configuration of $m$ points in $\Bbb C^n$, we give a closed formula of the algebraic complexity of the hypothesis variety as $m$ grows for the case of $n=1$. For the case $n>1$ we conjecture a generalization of this formula supported by numerical experiments.