Testing the variety hypothesis
Algebraic Geometry
2025-07-23 v1 Metric Geometry
Statistics Theory
Machine Learning
Statistics Theory
Abstract
Given a probability measure on the unit disk, we study the problem of deciding whether, for some threshold probability, this measure is supported near a real algebraic variety of given dimension and bounded degree. We call this "testing the variety hypothesis". We prove an upper bound on the so-called "sample complexity" of this problem and show how it can be reduced to a semialgebraic decision problem. This is done by studying in a quantitative way the Hausdorff geometry of the space of real algebraic varieties of a given dimension and degree.
Cite
@article{arxiv.2507.16705,
title = {Testing the variety hypothesis},
author = {A. Lerario and P. Roos Hoefgeest and M. Scolamiero and A. Tamai},
journal= {arXiv preprint arXiv:2507.16705},
year = {2025}
}