Quotient Geometry and Persistence-Stable Metrics for Swarm Configurations

Abstract

Swarm and constellation reconfiguration can be viewed as motion of an unordered point configuration in an ambient space. Here, we provide persistence-stable, symmetry-invariant geometric representations for comparing and monitoring multi-agent configuration data. We introduce a quotient formation space S_n(M,G)=M^n/(G S_n) and a formation matching metric d_M,G obtained by optimizing a worst-case assignment error over ambient symmetries g G and relabelings σ S_n. This metric is a structured, physically interpretable relaxation of Gromov--Hausdorff distance: the induced inter-agent metric spaces satisfy d_GH(X_x,X_y) d_M,G([x],[y]). Composing this bound with stability of Vietoris--Rips persistence yields d_B(Φ_k([x]),Φ_k([y])) d_M,G([x],[y]), providing persistence-stable signatures for reconfiguration monitoring. We analyze the metric geometry of (S_n(M,G),d_M,G): under compactness/completeness assumptions on M and compact G it is compact/complete and the metric induces the quotient topology; if M is geodesic then the quotient is geodesic and exhibits stratified singularities along collision and symmetry strata, relating it to classical configuration spaces. We study expressivity of the signatures, identifying symmetry-mismatch and persistence-compression mechanisms for non-injectivity. Finally, in a phase-circle model we prove a conditional inverse theorem: under semicircle support and a gap-labeling margin, the H_0 signature is locally bi-Lipschitz to d_M,G up to an explicit factor, yielding two-sided control. Examples on S^2 and T^m illustrate satellite-constellation and formation settings.

Reproductions