论文标题
通过低级近似追踪完成张量
Tensor Completion via a Low-Rank Approximation Pursuit
论文作者
论文摘要
本文考虑了有限采样的张量（也称为多维阵列）的张量（也称为多维阵列）的完成问题。我们的贪婪方法是基于扩展矩阵完成的低级别近似追踪（LRAP）方法以完成张量。该方法使用张量奇异值分解（T-SVD）进行张量分解，该分解值将标准矩阵SVD扩展到张量。 T-SVD导致等级的概念，称为Tubal-Rank。我们希望在这里可以从低分辨率样本中重新创建张量的数据。为了成功完成低分辨率张量，我们假设给定的张量数据量低。对于低管数组的张量，我们为基于张量限制的等轴测特性（TRIP）建立了收敛结果。我们的张量跳闸条件的结果类似于在RIP条件下的低级矩阵完成。 Trip条件使用T-SVD进行低管量张量，而RIP则将SVD用于矩阵。我们表明，Subgaussian测量图以很高的概率满足了行程条件，并且几乎可以在所需的测量数量上获得最佳结合。我们将所提出算法的数值性能与用于视频恢复和颜色图像恢复的最新方法的数值性能。
This paper considers the completion problem for a tensor (also referred to as a multidimensional array) from limited sampling. Our greedy method is based on extending the low-rank approximation pursuit (LRAP) method for matrix completions to tensor completions. The method performs a tensor factorization using the tensor singular value decomposition (t-SVD) which extends the standard matrix SVD to tensors. The t-SVD leads to a notion of rank, called tubal-rank here. We want to recreate the data in tensors from low resolution samples as best we can here. To complete a low resolution tensor successfully we assume that the given tensor data has low tubal-rank. For tensors of low tubal-rank, we establish convergence results for our method that are based on the tensor restricted isometry property (TRIP). Our result with the TRIP condition for tensors is similar to low-rank matrix completions under the RIP condition. The TRIP condition uses the t-SVD for low tubal-rank tensors, while RIP uses the SVD for matrices. We show that a subgaussian measurement map satisfies the TRIP condition with high probability and gives an almost optimal bound on the number of required measurements. We compare the numerical performance of the proposed algorithm with those for state-of-the-art approaches on video recovery and color image recovery.
