标题： 低管秩张量逼近的高效算法及其应用 显示英文标题

标题： Efficient Algorithms for Low Tubal Rank Tensor Approximation with Applications

摘要： 在本文中，我们提出了高效的随机固定精度技术，用于张量的低管秩近似。所提出的方法比现有的固定精度算法在近似截断张量奇异值分解（T-SVD）方面更快且更高效。此外，有一些关于计算张量低管秩近似随机单次遍历算法的研究，但它们都没有实验性地报告此类算法在真实世界数据张量（例如图像和视频）的低秩近似中的鲁棒性。当前的单次遍历算法是针对矩阵开发的算法在张量上的推广。然而，最近矩阵的单次遍历随机算法已经得到了改进和稳定。受这一进展的启发，本文基于管积（T-乘积）将这些算法推广到张量情况。我们进行了广泛的模拟以研究它们与现有单次遍历随机算法的鲁棒性。特别是，我们实验发现，具有相等大小的抽样参数的单次遍历算法通常会导致病态张量最小二乘问题和不准确的结果。实验表明，我们提出的单次遍历算法在这一意义上是鲁棒的。数值结果表明，在相同条件下（设置相同的超参数），我们提出的算法表现更好。还展示了三个应用：图像压缩、超分辨率问题和深度学习。 显示英文摘要

摘要： In this paper we propose efficient randomized fixed-precision techniques for low tubal rank approximation of tensors. The proposed methods are faster and more efficient than the existing fixed-precision algorithms for approximating the truncated tensor SVD (T-SVD). Besides, there are a few works on randomized single-pass algorithms for computing low tubal rank approximation of tensors, none of them experimentally reports the robustness of such algorithms for low-rank approximation of real-world data tensors e.g., images and videos. The current single-pass algorithms for tensors are generalizations of those developed for matrices to tensors. However, the single-pass randomized algorithms for matrices have been recently improved and stabilized. Motivated by this progress, in this paper, we also generalize them to the tensor case based on the tubal product (T-product). We conduct extensive simulations to study the robustness of them compared with the existing single-pass randomized algorithms. In particular, we experimentally found that single-pass algorithms with the sketching parameters of equal sizes usually lead to ill-conditioned tensor least-squares problems and inaccurate results. It is experimentally shown that our proposed single-pass algorithms are robust in this sense. Numerical results demonstrate that under the same conditions (setting the same hyper-parameters), our proposed algorithms provide better performance. Three applications to image compression, super-resolution problem and deep learning are also presented.