Given an inner product space $V$ and a group $G$ of linear isometries, max filtering offers a rich class of convex $G$-invariant maps. In this paper, we identify sufficient conditions under which these maps are locally bilipschitz on $R(G)$, the set of orbits with maximal dimension, with respect to the quotient metric on the orbit space $V/G$. Central to our proof is a desingularization theorem, which applies to open, dense neighborhoods around each orbit in $R(G)/G$ and may be of independent interest. As an application, we provide guarantees for stable weighted phase retrieval. That is, we construct componentwise convex bilipschitz embeddings of weighted complex (resp.\ quaternionic) projective spaces. These spaces arise as quotients of direct sums of nontrivial unitary irreducible complex (resp.\ quaternionic) representations of the group of unit complex numbers $S^1\cong \operatorname{SO}(2)$ (resp.\ unit quaternions $S^3\cong \operatorname{SU}(2)$). We also discuss the relevance of such embeddings to a nearest-neighbor problem in single-particle cryogenic electron microscopy (cryo-EM), a leading technique for resolving the spatial structure of biological molecules.

翻译：给定内积空间$V$与线性等距群$G$，最大滤波提供了一类丰富的凸$G$不变映射。本文确定了在轨道空间$V/G$的商度量下，这些映射在最大维轨道集合$R(G)$上局部双利普希茨连续的充分条件。证明的核心是一个去奇性定理，该定理适用于$R(G)/G$中每个轨道周围的稠密开邻域，可能具有独立的理论价值。作为应用，我们为稳定加权相位恢复提供了理论保证。具体而言，我们构造了加权复（或四元数）射影空间的分量凸双利普希茨嵌入。这些空间源于单位复数群$S^1\cong \operatorname{SO}(2)$（或单位四元数群$S^3\cong \operatorname{SU}(2)$）的非平凡酉不可约复（或四元数）表示直和的商空间。本文还探讨了此类嵌入在单颗粒冷冻电子显微镜（冷冻电镜）最近邻问题中的潜在应用，该技术是解析生物分子空间结构的主流方法。