We develop a framework for learning dependence structures from empirical dependence operators. Rather than treating cluster, factor, and sparse dependence as maintained assumptions, we represent them as covariance geometries in a common Hilbert space and summarize dependence through a low-dimensional dependence profile based on projection similarity scores. We establish identification under a principal-angle separation condition, prove consistency and asymptotic normality of the estimated profile, and derive finite-sample classification error bounds. We further show that when covariance-geometry tangent spaces overlap, no statistical procedure can distinguish the geometries at first order, providing a formal characterization of ambiguous dependence structures. Projection-residual diagnostics assess absolute goodness-of-fit and detect misspecified covariance dictionaries. Finally, we establish oracle adaptivity of profile-guided inference: dependence profiles can be used to select dependence-robust procedures in a data-driven manner, yielding inference that is asymptotically equivalent to an infeasible oracle that knows the dominant covariance geometry in advance.

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