Local Depth-Based Corrections to Maxmin Landmark Selection for Lazy Witness Persistence

TL;DR

This paper introduces local depth-based corrections to maxmin landmark selection for lazy witness persistence, providing mathematical guarantees and empirical evidence of improved geometric properties.

Contribution

It presents a novel local depth-based correction method for maxmin landmark selection, with mathematical proofs and extensive empirical validation.

Findings

Support-weighted partial recentering improves geometric robustness over maxmin.

The method maintains topological summaries while enhancing geometric coverage.

Empirical results show consistent improvements in 2D benchmarks, with mixed results in 3D.

Abstract

We study a family of local depth-based corrections to maxmin landmark selection for lazy witness persistence. Starting from maxmin seeds, we partition the cloud into nearest-seed cells and replace or move each seed toward a deep representative of its cell. The principal implemented variant, \emph{support-weighted partial recentering}, scales the amount of movement by cell support. The contributions are both mathematical and algorithmic. On the mathematical side, we prove local geometric guarantees for these corrections: a convex-core robustness lemma derived from halfspace depth, a $2r$ cover bound for subset recentering, and projected cover bounds for the implemented partial-recentering rules. On the algorithmic side, we identify a practically effective variant through a layered empirical study consisting of planar synthetic benchmarks, a parameter-sensitivity study, and an MPEG-7 silhouette benchmark, together with a modest three-dimensional torus extension. The main planar experiments show that support-weighted partial recentering gives a consistent geometric improvement over maxmin while preserving the thresholded $H_1$ summary used in the study. The three-dimensional experiment shows the same geometric tendency but only mixed topological behavior. The paper should therefore be read as a controlled study of a local depth-based alternative to maxmin, rather than as a global witness-approximation theorem or a claim of uniform empirical superiority.