Tensor Principal Component Analysis in High Dimensional CP Models
Tensor Principal Component Analysis in High Dimensional CP Models
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DOI：
10.1109/tit.2022.3203972
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发表时间：
2021-08
影响因子：
2.5
通讯作者：
Yuefeng Han;Cun-Hui Zhang
中科院分区：
文献类型：
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作者：
Yuefeng Han;Cun-Hui Zhang
高维非正交尖张量的CP分解是一个具有广泛应用的重要问题。然而，以往有理论保证的工作通常在CP分量的基向量上假设了限制性非相干条件。在本文中，我们提出了新的计算效率高的复合PCA和并发正交化算法，用于在轻度相干条件下具有理论保证的张量CP分解。复合主成分分析将主成分分解或奇异值分解两次，首先对张量数据进行矩阵展开得到奇异向量，然后对第一步得到的奇异向量进行矩阵折叠。它可以作为张量CP分解的任何迭代优化方案的初始化。并发正交化算法通过对其他模式下其他CP分量生成的空间的正交补同时施加投影来迭代估计张量的每个模式下的基向量。它的目的是改进低或中高CP阶张量的交替最小二乘估计和其他形式的高阶正交迭代，并保证任意给定初始估计的误差在一个小常数范围内具有二阶或更高阶收敛性。我们的理论研究提供了两种算法的估计精度和收敛速度。这两种算法都适用于确定性张量及其噪声版本，以及具有不相关因素的因子模型中阶K张量数据的阶2K协方差张量。仿真实验证明了我们的方法比现有方法具有显著的实用优势。
The CP decomposition for high dimensional non-orthogonal spiked 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 other 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, and it is guaranteed to have second or higher order convergence when the error of any given initial estimator is bounded by a small constant. Our theoretical investigation provides estimation accuracy and convergence rates for the two proposed algorithms. Both proposed algorithms are applicable to deterministic tensor, its noisy version, and the order- $2K$ covariance tensor of order- $K$ tensor data in a factor model with uncorrelated factors. Simulation experiments demonstrate significant practical superiority of our approach over existing methods.