Regularity Conditions for Critical Point Convergence

TL;DR

This paper investigates conditions under which the topological features of a sequence of functions on a manifold converge to those of a limit function, with implications for data-derived models in empirical sciences.

Contribution

It establishes sufficient regularity conditions for the convergence of critical points and topological features of functions in $C^1$ and $C^2$ settings, using advanced mathematical tools.

Findings

Conditions for convergence of local maxima, minima, and saddle points.

Application of Poincaré-Hopf and mountain pass theorems in $C^1$ setting.

Reformulation of results in empirical process language.

Abstract

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.