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We develop a novel theoretical framework for understating OT schemesrespecting a class structure. For this purpose, we propose a convex OT programwith a sum-of-norms regularization term, which provably recovers the underlyingclass structure under geometric assumptions. Furthermore, we derive anaccelerated proximal algorithm with a closed-form projection and proximaloperator scheme, thereby affording a more scalable algorithm for computingoptimal transport plans. We provide a novel argument for the uniqueness of theoptimum even in the absence of strong convexity. Our experiments show that thenew regularizer not only results in a better preservation of the classstructure in the data but also yields additional robustness to the datageometry, compared to previous regularizers.

Recovery Bounds on Class-Based Optimal Transport: A Sum-of-Norms Regularization Framework