Low rank tensor approximation is an important subject with a wide range of applications. Most prevailing techniques for computing the low rank approximation in the Tucker format often first assemble relevant factors into matrices and then update by turns one factor matrix at a time. In order to improve two factor matrices simultaneously, a special system of nonlinear matrix equations over a certain product Stiefel manifold must be resolved at every update. The solution to the system consists of orbit varieties invariant under the orthogonal group action, which thus imposes challenges on its analysis. This paper proposes a scheme similar to the power method for subspace iterations except that the polar decomposition is used as the normalization process and that the iteration can be applied to both the orbits and the cross-sections. The notion of quotient manifold is employed to factor out the effect of orbital solutions. The dynamics of the iteration is completely characterized. An isometric isomorphism between the tangent spaces of two properly identified Riemannian manifolds is established to lend a hand to the proof of convergence.

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极坐标分解的非线性类幂迭代及其在张量逼近中的应用

低秩张量近似是一门具有广泛应用的重要课题。用于计算 Tucker 格式中的低秩近似的大多数流行技术通常首先将相关因子组装成矩阵，然后一次轮流更新一个因子矩阵。为了同时改进两个因子矩阵，必须在每次更新时求解特定积 Stiefel 流形上的非线性矩阵方程的特殊系统。该系统的解由正交群作用下不变的轨道变体组成，因此对其分析提出了挑战。本文提出了一种类似于幂法的子空间迭代方案，不同之处在于使用极坐标分解作为归一化过程，并且迭代可以应用于轨道和横截面。商流形的概念用于分解轨道解的影响。迭代的动态是完全表征的。建立了两个正确识别的黎曼流形的切空间之间的等距同构，以帮助证明收敛。