Circular Optimal Partial Transport

When two signals share only part of their content, optimal partial transport (OPT) can match that common content without assigning every observation a counterpart. For periodic signals, the samples lie on a circle, and their cyclic ordering requires more than a direct use of a line solver. PAWC computes the complete circular partial-transport profile for the $W_1$ cost on distinct support. Building on the PAWL and PAWC profile algorithms, we introduce COPT, an exact solver for circular OPT between empirical measures with a penalty on unmatched mass, an arbitrary convex radial cost, and possibly repeated support points. We extend PAWC's local augmentation and simultaneous-cut arguments to convex costs, using a rank refinement to handle repeated support and flat cost regions. We prove that COPT returns the complete profile of optimal costs over all transported cardinalities in $O(N\log N+K^2)$ time and $O(N)$ memory, where $N$ is the number of atoms and $K$ the largest transported cardinality, and in $O(pN{\log }^{2}N)$ time for integer power costs. We further extend COPT to spherical measures through great-circle slicing and to repeated comparisons through a linear embedding, and we establish their metric properties. Through numerical experiments on correspondence recovery, partial shape matching, foreground retrieval, and directional data, we demonstrate the computational benefits of COPT and its robustness to missing signal and background clutter.