{"ID":2891876,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.16705","arxiv_id":"2507.16705","title":"Testing the variety hypothesis","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.","short_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 complexit...","url_abs":"https://arxiv.org/abs/2507.16705","url_pdf":"https://arxiv.org/pdf/2507.16705v1","authors":"[\"A. Lerario\",\"P. Roos Hoefgeest\",\"M. Scolamiero\",\"A. Tamai\"]","published":"2025-07-22T15:40:55Z","proceeding":"math.AG","tasks":"[\"math.AG\",\"math.MG\",\"math.ST\",\"stat.ML\"]","methods":"[]","has_code":false}