Title: Optimization on Multiple Manifolds. Abstract: Optimization on manifold has been widely used in machine learning, to handle optimization problems with constraint. Most previous works focus on the case with a single manifold. However, in practice it is quite common that the optimization problem involves more than one constraints, (each constraint corresponding to one manifold). It is not clear in general how to optimize on multiple manifolds effectively and provably especially when the intersection of multiple manifolds is not a manifold or cannot be easily calculated. We propose a unified algorithm framework to handle the optimization on multiple manifolds. Specifically,  we integrate information from multiple manifolds and move along an ensemble direction by viewing the information from each manifold as a drift and adding them together. We prove the convergence properties of the proposed algorithms. We also apply the algorithms into  training neural network with batch normalization layers and achieve preferable empirical results.