Riemannian Stochastic Variance-Reduced Cubic Regularized Newton Method for Submanifold Optimization

We propose a stochastic variance-reduced cubic regularized Newton algorithm to optimize the finite-sum problem over Riemannian submanifold of Euclidean space. The proposed requires full gradient and Hessian update at beginning each epoch while it performs updates in iterations within epoch. iteration complexity $$O(\epsilon ^{-3/2})$$ obtain an $$(\epsilon ,\sqrt{\epsilon })$$ -second-order stationary point, i.e., point with norm upper bounded by $$\epsilon $$ minimum eigenvalue lower $$-\sqrt{\epsilon }$$ , is established when manifold embedded Furthermore, paper proposes computationally more appealing modification which only inexact solution subproblem same complexity. evaluated compared three other second-order methods two numerical studies on estimating inverse scale matrix multivariate t-distribution symmetric positive definite matrices parameter linear classifier sphere manifold.