An Inexact Riemannian Proximal Momentum Variance-Reduced Method: Complexity Bounds and KL Sequential Convergence

We develop a unified analysis of inexact stochastic Riemannian proximal optimization for finite-sum nonsmooth composite problems over compact embedded submanifolds. The framework accommodates variance-reduced gradient estimators, projected momentum, and inexact tangent-space proximal solves under a single conditional error-dissipation condition, verified for projection-based SVRG, SARAH/SPIDER, SAGA, and SAG. A computable Fenchel-dual residual criterion, with tolerance prescribed before sampling and inner iterations, enables explicit control of the inner work. We establish conditional expected descent, subsequential stationarity, and an $O({\epsilon }^{-2})$ outer complexity. With SARAH/SPIDER and accumulative regularization, iRPMVR attains $O(n+\sqrt{n}{\epsilon }^{-2})$ component-gradient and $O({\epsilon }^{-3})$ proximal-operator complexities. We further develop an abstract KL principle for conditional expected descent with memory and summable tails using only the ordinary pointwise KL property. A counterexample shows that a power-type expected-KL implication used in earlier stochastic analyses can fail. The principle yields almost-sure finite length, whole-sequence convergence, and deterministic KL rates.