Testing the variety hypothesis

TL;DR

This paper investigates the problem of determining whether a probability measure on the unit disk is concentrated near a real algebraic variety of specified dimension and degree, providing bounds on the sample complexity for this testing task.

Contribution

It introduces a method to test the variety hypothesis by reducing it to a semialgebraic decision problem and analyzes the geometric complexity involved.

Findings

Established an upper bound on sample complexity for the testing problem

Reduced the testing problem to a semialgebraic decision problem

Analyzed the Hausdorff geometry of algebraic varieties space

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.