The CP decomposition for high dimensional non-orthogonal spike tensors is an important problem with broad applications across many disciplines. However, previous works with theoretical guarantee typically assume restrictive incoherence conditions on the basis vectors for the CP components. In this paper, we propose new computationally efficient composite PCA and concurrent orthogonalization algorithms for tensor CP decomposition with theoretical guarantees under mild incoherence conditions. The composite PCA applies the principal component or singular value decompositions twice, first to a matrix unfolding of the tensor data to obtain singular vectors and then to the matrix folding of the singular vectors obtained in the first step. It can be used as an initialization for any iterative optimization schemes for the tensor CP decomposition. The concurrent orthogonalization algorithm iteratively estimates the basis vector in each mode of the tensor by simultaneously applying projections to the orthogonal complements of the spaces generated by others CP components in other modes. It is designed to improve the alternating least squares estimator and other forms of the high order orthogonal iteration for tensors with low or moderately high CP ranks. Our theoretical investigation provides estimation accuracy and statistical convergence rates for the two proposed algorithms. Our implementations on synthetic data demonstrate significant practical superiority of our approach over existing methods.
翻译:高维非正统悬浮加压粒子的氯化石蜡分解是多个学科广泛应用的一个重要问题。然而,先前的理论保障工作通常假定在氯化石蜡组件的向量基矢量上存在限制性的不一致条件。在本文件中,我们提议采用新的计算高效复合五氯苯和同时的正压CP分解算法,并在轻度不一致性条件下以理论保证的方式对高压氯化石蜡分解进行新的计算高效复合五氯苯和同时的正对振动算法,同时将预测用于其他模式中氯化石蜡组件产生的空间的正方形或单值分解两次应用主要成分或单值分解。它旨在改进为获得单向矢量数据而展开的矩阵,然后是第一步获得的单向矢量向矢量单矢量的叠合矩阵。它可以用作任何调动式优化变色色色的调离子系统。同时,通过将预测用于其他模式中性CP组件生成的空间的或分解补充空间。它旨在改进最不相平方方形体的估测算器,以及第一步获得的单质或正向导变异矢量式矢量的矩阵矩阵的矩阵。同时,我们为目前对压压的理论级化的精确度进行中度,为我们现有的压压压压压压压压压压压压压压压压压压压压压压压压压的演算。