标题： 临界点收敛的正则条件 显示英文标题

标题： Regularity Conditions for Critical Point Convergence

摘要： 我们关注在紧致流形带边界 $S$上定义的函数序列 $\{f_n\}$，它们在 $C^k$度量下收敛到一个极限 $f$。 在实证科学中，一个常见的隐含假设是，当这些函数表示从数据中得出的随机过程时， $f_n$的拓扑特征最终将类似于 $f$的拓扑特征。 在这项工作中，我们在各种正则性假设下研究该说法的有效性，目标是找到使得此类函数的局部极大值、极小值和鞍点数量收敛的充分条件。 在 $C^1$ 设置中，我们通过采用较少为人知的 Poincaré-Hopf 和山路定理的变体来实现这一点，在 $C^2$ 设置中，我们采用一种受基于同伦的 Morse 引理证明启发的方法。 为了便于实际应用，我们最后将我们的中心定理用经验过程的语言重新表述。 显示英文摘要

摘要： We focus on a sequence of functions $\{f_n\}$, defined on a compact manifold with boundary $S$, converging in the $C^k$ metric to a limit $f$. A common assumption implicitly made in the empirical sciences is that when such functions represent random processes derived from data, the topological features of $f_n$ will eventually resemble those of $f$. In this work, we investigate the validity of this claim under various regularity assumptions, with the goal of finding conditions sufficient for the number of local maxima, minima and saddle of such functions to converge. In the $C^1$ setting, we do so by employing lesser-known variants of the Poincar\'{e}-Hopf and mountain pass theorems, and in the $C^2$ setting we pursue an approach inspired by the homotopy-based proof of the Morse Lemma. To aid practical use, we end by reformulating our central theorems in the language of the empirical processes.