AI 中文总结
本文针对高维实空间中概率分布是否集中在实代数完全交附近的问题，设计了明确有效的学习与检验算法，基于正则多项式系统的定量几何估计，确定了算法的样本与算术复杂度界。
AI 中文摘要
给定来自潜在高维实空间中概率分布的独立同分布样本，我们研究该分布是否集中在具有指定维数、有界次数和有界条件数的实代数完全交附近的检验问题。我们设计了一种明确且有效的学习过程，该过程要么在近似阈值的可控松弛范围内证明此类流形不存在，要么返回具有可控几何复杂度的候选回归流形；等价地，该过程在该假设类内检验流形假设。所提出的过程依赖于正则多项式系统的定量几何估计，这会产生一个易处理的辅助优化问题。随后，我们开发了一种数据驱动算法来求解该辅助优化问题，并确定了其样本复杂度和算术复杂度的明确界。
英文摘要
Given independent and identically distributed samples samples from a probability distribution in a potentially high-dimensional real space, we study the problem of testing whether the distribution is concentrated near a real algebraic complete intersection of prescribed dimension, bounded degree, and bounded condition number. We design an explicit and effective learning procedure which either certifies the nonexistence of such a manifold, up to a controlled relaxation of the approximation threshold, or returns a candidate regression manifold with controlled geometric complexity. Equivalently, the procedure tests the manifold hypothesis within this hypothesis class. The proposed procedure relies on quantitative geometric estimates for regular polynomial systems, which lead to a tractable auxiliary optimization problem. We then develop a data-driven algorithm to solve this auxiliary optimization problem, establishing explicit bounds on its sample and arithmetic complexity.
