Operator-Specific Information Retention in Spectral Membrane Analysis

Abstract

Membrane analysis in cryo-electron tomography proceeds through representation changes — meshing, graph construction, operator construction, clipping, decimation, and spectral truncation — that are not information-neutral. Graph-based and cotangent geometry-aware descriptors often diverge under transformation, especially in morphologies with constrictions, branching, or sheet-like structure. This paper interprets such divergences as measurable signals of operator-specific information retention rather than nuisance variation.We introduce a framework decomposing membrane information into structural components (connectivity, bottlenecks, low-mode organization) and distributional components (degree and curvature entropy), then evaluate their behavior under specified operators. We define retention, deformation, and disagreement functionals and instantiate them using normalized graph algebraic connectivity, normalized cotangent second eigenvalue, and two entropy descriptors. The framework is evaluated on five synthetic membrane families and a real-membrane bonus set (IMM, OMM, ER from EMPIAR-11370).Three patterns recur. First, baseline morphology already separates topology-like and geometry-aware channels, especially in constricted geometries. Second, radial clipping produces morphology-dependent structural responses typically larger than distributional shifts. Third, graph-cotangent disagreement under transformation is largest where morphology makes the representation most informative. Decimation with largest-connected-component cleanup reveals three regimes — clean simplification, transition, and fragmentation stress — showing that aggressive simplification changes the topology of the analysis object rather than merely coarsening it. Spectral truncation indicates that cotangent diffusion structure concentrates in low modes.The resulting picture is methodological: membrane descriptors do not respond uniformly to representation change, and the divergence between topology-like and geometry-aware channels is often the signal rather than the noise.

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