Intersection-aware Tangent Decomposition for Manifold Clustering

Clustering intersecting manifolds requires distinguishing nearby observations whose generating branches have different tangent directions. We study when local branch representations can make such clustering possible, and present Intersection- aware Tangent-decomposition Clustering (ITC) as a practical construction mo- tivated by that analysis. The abstract theory identifies sufficient conditions for recovery: a local residual margin controls branch assignment, and a graph perturba- tion bound tolerates arbitrary tangent errors on a vanishing fraction of observations. For balanced orthogonal line segments, we establish a nonempty fixed-radius regime in which imperfect representations and asymptotically optimal two-means rounding yield vanishing misclassification error. This provides an existence re- sult for recoverable intersecting-manifold configurations, together with stability beyond exact tangents. A small-angle oracle example shows why graph recov- erability is a separate requirement. The practical method fits affine branches at spatially distributed anchors, assigns one projector per observation, and constructs a radius-supported affinity. Its finite fitting and spectral routines provide a concrete implementation of the design principles, evaluated empirically; the existence guar- antee concerns the stated abstract realization. A further conditional result covers the degree-rescaled embedding used by the backend.