发表机构
Institute of Basic Sciences, Seoul National University; Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology; Department of Statistics, Seoul National University（首尔大学基础科学研究院; 韩国科学技术院数学科学系; 首尔大学统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究拓扑数据分析中持久强度函数的双样本检验，提出基于核的置换检验，通过引入假设控制持久图基数影响，给出方差界，证明概率模型广泛性，建立极小极大最优性，模拟和实际应用验证了方法有效性和高功效。
AI 中文摘要
强度函数被定义为持久图预期测度的勒贝格密度，是拓扑数据分析中持久图概率分布的基本总结。尽管已有多种估计强度函数的方法，但强度函数的统计假设检验仍未充分探索。我们提出基于核的置换检验，并分析其针对持久强度函数差异特征的备择假设的功效。引入控制持久图可能无界基数影响的假设，为检验统计量给出精确方差界。还表明概率模型广泛，能包含\(\mathbb{R}^2\)中\(y > x \geq 0\)子集上的所有概率密度，借此建立所提检验的极小极大最优性。推导了圆上Čech复形持久图的显式特征。因实际中最优带宽不可直接获取，采用带宽聚合框架。模拟和实际数据应用证明了有效性和高经验功效。
英文摘要
Persistence intensity functions provide interpretable and informative first-order summaries of random persistence diagram distributions. We study two-sample testing for equality of persistence intensity functions, allowing the underlying diagram distributions to differ under the null. We construct a weighted-kernel statistic as an unbiased estimator of the squared reproducing-kernel Hilbert space distance between the corresponding weighted intensity embeddings, and calibrate it by studentization. For shrinking bandwidths, we establish uniform asymptotic normality under the null and thereby obtain asymptotic Type I error control. Its power is characterized in terms of the $L^2$ discrepancy between the weighted intensity functions. To accommodate persistence diagrams with possibly unbounded cardinality, we introduce regularity conditions that control the effect of cardinality variation and yield the desired moment bounds for the test statistic. We further show that every probability density on $\{(x,y)\in\mathbb{R}^2:y>x > 0\}$ can be realized as the persistence intensity function of a random diagram. Using these results, we establish that the proposed test attains minimax-optimal separation rates over anisotropic Sobolev balls. Lastly, since the optimal bandwidth is not directly accessible in practice, we adapt a bandwidth aggregation framework.
