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.

翻译：我们研究了一族基于局部深度的修正方法，用于懒惰见证持久性中的最大最小地标选择。从最大最小种子出发，我们将点云划分为最近种子单元，并将每个种子替换或移动至其单元的一个深度代表点。主要实现的变体——\emph{支持加权部分重定位}——通过单元支持度来缩放移动量。本文的贡献兼具数学与算法两方面。在数学层面，我们证明了这些修正的局部几何保证：基于半空间深度的凸核鲁棒性引理、子集重定位的$2r$覆盖界，以及已实现部分重定位规则的投影覆盖界。在算法层面，我们通过分层实证研究（包括平面合成基准测试、参数敏感性研究、MPEG-7轮廓基准测试以及一个适度的三维环面扩展）确定了一个实际有效的变体。主要平面实验表明，支持加权部分重定位在保持研究中使用的阈值化$H_1$摘要的同时，相较于最大最小方法实现了持续的几何改进。三维实验显示出相同的几何趋势，但仅表现出混合的拓扑行为。因此，本文应被视为对最大最小方法的一种局部深度替代方案的控制性研究，而非全局见证逼近定理或统一实证优越性的主张。