CLASP: Elastic Shape Clustering of Closed Curves

We introduce CLASP, a novel unified framework for cluster analysis of closed curves, functional data that trace a loop and return to their starting point, arising in applications such as vectorcardiography, cardiac imaging, and spirometry. Our framework features an ambient Gauss-Newton retraction onto the closure manifold, for which we derive the Gram matrix in closed form; an FFT-based seed search that provably confines the optimal starting point to an explicit candidate set at a fraction of the cost of exhaustive search; and a provably invariant amplitude-phase decomposition that separates the geometry of a curve from its timing. Additionally, we incorporate a compositional clustering layer that combines nine standard feature channels and fuses them without labels through two complementary weighted consensus paths, arbitrated into a single partition using fixed default hyperparameters throughout. We establish the universal approximation property of closure-respecting decoders built on the retraction, and validate the effectiveness of our framework through extensive experiments on simulated designs, shape-outline benchmarks, and clinical closed-loop datasets.