Regularity Conditions for Critical Point Convergence

2025/07/02 by Thomas J. Maullin-Sapey, Maullin-Sapey, Thomas J., Samuel Davenport +1
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #General Topology (math.GN) #Probability (math.PR) #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2507.01854

openalex publication_date 2025/07/02 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

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 C¹ setting, we do so by employing lesser-known variants of the Poincaré-Hopf and mountain pass theorems, and in the C² 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.

Citations

• Object detection under the linear subspace model with application to cryo-EM images
• Confidence regions for the location of peaks of a smooth random field
• Spatial Confidence Regions for Combinations of Excursion Sets in Image Analysis
• Introduction to Random Fields
• Multiple Testing of Local Maxima for Detection of Peaks in Random Fields
• A generalized Poincaré-Hopf index theorem

Related

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• Topology and convergence on the space of measure-valued functions