A max filtering local stability theorem with application to weighted phase retrieval and cryo-EM

Abstract: 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 lower Lipschitz 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.