# Testing the variety hypothesis

arXiv: https://arxiv.org/abs/2507.16705 · HTML: https://mlsanity.com/paper/2507.16705/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 16 references that are in the map.
- First posted on arXiv 22 Jul 2025.

## Builds on (most cited first, 3 of 3)

- [The Euclidean distance degree of an algebraic variety](https://mlsanity.com/paper/1309.0049/) (2013)
- [Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applications](https://mlsanity.com/paper/2311.05116/) (2023)
- [Hausdorff approximations and volume of tubes of singular algebraic sets](https://mlsanity.com/paper/2104.05053/) (2021)

## Built on by (most cited first, 1 of 1)

- [Testing Algebraic Complete Intersections](https://mlsanity.com/paper/2610.12288/) (2026)

## Abstract (arXiv)

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.