We consider the problem of learning low-rank tensors from partial observations with structural constraints, and propose a novel factorization of such tensors, which leads to a simpler optimization problem. The resulting problem is an optimization problem on manifolds. We develop first-order and second-order Riemannian optimization algorithms to solve it. The duality gap for the resulting problem is derived, and we experimentally verify the correctness of the proposed algorithm. We demonstrate the algorithm on nonnegative constraints and Hankel constraints.

翻译：我们考虑从部分观测中学习具有结构约束的低秩张量问题，并提出一种新型张量分解方法，该方法简化了优化问题。最终问题转化为流形上的优化问题。我们开发了一阶和二阶黎曼优化算法进行求解，推导了该问题的对偶间隙，并通过实验验证了所提算法的正确性。我们分别在非负约束和汉克尔约束下演示了该算法的性能。