Comparing dynamical operators with flags

Many scientific and engineering domains, including neuroscience, fluid dynamics, mechanics, and climate science, study ensembles of multidimensional spatiotemporal fields through spectral or modal decompositions. A recurring challenge is to compare such decompositions across many realizations while respecting both their intrinsic geometric characteristics and their hierarchy of dynamical timescales. Euclidean distances are common across these disciplines, but they are sensitive to arbitrary basis choices, whereas representing all modes by a single subspace discards their timescale organization. We represent each realization by a flag: a nested sequence of subspaces formed by grouping dynamical modes by spectral conjugacy and ordering them by decay time. We equip the space of flags with a chordal distance that lets us compare decompositions across realizations. The resulting distance is gauge-invariant by construction; we prove that it defines a metric, admits an isometric Euclidean embedding (thus enabling standard kernel methods), and is stable under perturbations of the fitted operator away from decay-time crossings. We illustrate the approach in climate science on 216 realizations from four CMIP6 models and a reanalysis across three variables, as well as synthetic ensembles with known ground truth. The flag geometry separates dynamical structures that the Euclidean baseline fails to distinguish and regularly outperforms a flat-subspace representation