基于莫尔斯理论、持久性和非参数估计的统计拓扑
Statistical topology via Morse theory, persistence and nonparametric estimation
摘要：
在本文中，我们将探讨多元统计的拓扑方法的使用。利用计算代数拓扑学中的持久同调，利用随机样本构造持久同调的估计。然后可以使用估计的持久同源性和真实持久同源性之间的瓶颈距离来评估该估计过程。与统计学的联系来自这样一个事实，即当被视为非参数回归问题时，瓶颈距离由超范数损失限制。从而在Holder函数类上的超范数风险下，确定了流形上非参数回归问题的一个尖锐的渐近Minimax界.这为持久同源性估计在预期瓶颈距离方面提供了良好的收敛性质。
英文摘要：
In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators of persistent homology. This estimation procedure can then be evaluated using the bottleneck distance between the estimated persistent homology and the true persistent homology. The connection to statistics comes from the fact that when viewed as a nonparametric regression problem, the bottleneck distance is bounded by the sup-norm loss. Consequently, a sharp asymptotic minimax bound is determined under the sup-norm risk over Holder classes of functions for the nonparametric regression problem on manifolds. This provides good convergence properties for the persistent homology estimator in terms of the expected bottleneck distance.
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