标题： 测试多样性假设 显示英文标题

标题： Testing the variety hypothesis

摘要： 给定单位圆盘上的一个概率测度，我们研究的问题是，对于某个阈值概率，该测度是否集中在给定维数和有界次数的实代数簇附近。 我们称此为“检验代数簇假设”。 我们证明了这个问题所谓的“样本复杂度”的上界，并展示了如何将其简化为一个半代数决策问题。 这是通过定量地研究给定维数和次数的实代数簇空间的Hausdorff几何来完成的。 显示英文摘要

摘要： 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.