Proximal DCA for Fr\'echet Regression on Riemannian Manifolds with Bounded Curvature

Abstract

Fr\'echet regression generalizes linear regression to metric-space-valued responses by defining fitted values as minimizers of weighted Fr\'echet functionals. Since these weights may have mixed signs, the resulting objective is a signed barycenter problem rather than a standard convex barycenter problem. On Riemannian manifolds, this is further complicated by the lack of global geodesic convexity and possible nonsmoothness of squared distances near cut loci. We study signed Fr\'echet regression on complete manifolds with two-sided bounded sectional curvature. By restricting optimization to a strongly convex normal ball containing the response support, we use local smoothness, Hessian comparison, and Jacobi-field estimates to formulate the problem as a locally controlled Riemannian proximal DC problem. This leads to FRIDA (Fr\'echet Regression via Riemannian Iterative DC Algorithm), an exact and inexact proximal DC algorithm for computing regression fits. We prove existence and interiority of minimizers under explicit signed-weight conditions, establish curvature-dependent strong convexity of the proximal subproblems, and show descent and convergence of the iterates to stationary points. We also derive sublinear complexity estimates and, under real-analyticity, obtain full-sequence convergence with KL-type local rates. These results provide a rigorous optimization framework for signed Fr\'echet regression on manifolds with bounded curvature.