Inverse-Free Second-Order Momentum for Decentralized Riemannian Stochastic Optimization on Compact Submanifolds

We study decentralized stochastic optimization on compact embedded submanifolds and propose decentralized projected Riemannian stochastic Hessian-corrected momentum (DPRSHCM). The method pairs a correction for the full ambient variation of the Riemannian gradient with a projected iterate difference, yielding a quadratic remainder as tangent spaces change. It combines this correction with gradient tracking and metric projection, using one stochastic gradient and one Hessian–vector product per node per iteration. With a diminishing stepsize and momentum parameters independent of the iteration budget, DPRSHCM achieves $\tilde{\mathcal{O}}({ϵ}^{-3/2})$ per-node oracle complexity for reducing both the expected squared Riemannian gradient norm and mean squared consensus error below $ϵ$. A constant stepsize chosen for a prescribed budget removes the logarithmic factor. With exact local gradients and stochastic Hessian–vector products, fixed parameters achieve the same joint criterion in $\mathcal{O}({ϵ}^{-1})$ iterations. Experiments on PCA and matrix completion demonstrate the method's effectiveness and the computational benefits of reusing gradient computations in matrix completion.