FRIDA: Curvature-Controlled DC Optimization for Signed Fr\'echet Regression

Fr\'echet regression extends classical regression to metric-space-valued responses by minimizing weighted sums of squared distances. However, these weights may be negative, resulting in an optimization problem that yields a signed barycenter problem and lacks the same existence and convexity guarantees as convex combination-based barycenters. We study this problem on complete Riemannian manifolds with two-sided sectional curvature bounds and develop FRIDA, a proximal Riemannian difference-of-convex (DC) method with exact and inexact inner loops. On a strongly convex normal ball containing the responses, curvature comparison provides local smoothness estimates for the signed objective and convexity estimates for the FRIDA subproblems. Under explicit conditions on the signed weights, we establish the existence of interior minimizers and the well-posedness of each FRIDA subproblem. We further prove sufficient descent, stationarity of every accumulation point, and an $O((N+1)^{-1/2})$ bound on the smallest successive-step distance. When the manifold and objective are real analytic, the entire sequence converges to a stationary point at a rate determined by the KL exponent.