论文标题
流张量火车近似
Streaming Tensor Train Approximation
论文作者
论文摘要
张量火车是一种多功能工具，可压缩和使用高维数据和功能。在这项工作中，我们介绍了流量张量火车近似（STTA），这是一种新的算法，用于以张量列车格式近似给定的张量$ \ MATHCAL T $。 STTA通过原始数据的双面随机素描访问$ \ Mathcal t $，使其可以流且易于实现 - 与现有的确定性和随机张量训练式火车近似不同。该属性还允许STTA方便地利用$ \ Mathcal t $，例如稀疏性和各种低级张量格式以及其线性组合。当使用高斯随机矩阵进行素描时，STTA是可以接受的，该分析可以构建并扩展到矩阵的广义NyStröm近似值上的现有结果。我们的结果表明，如果适当选择草图的大小，则可以预期STTA将达到几乎最佳的近似误差。一系列数值实验说明了与现有的确定性和随机方法相比，STTA的性能。
Tensor trains are a versatile tool to compress and work with high-dimensional data and functions. In this work we introduce the Streaming Tensor Train Approximation (STTA), a new class of algorithms for approximating a given tensor $\mathcal T$ in the tensor train format. STTA accesses $\mathcal T$ exclusively via two-sided random sketches of the original data, making it streamable and easy to implement in parallel -- unlike existing deterministic and randomized tensor train approximations. This property also allows STTA to conveniently leverage structure in $\mathcal T$, such as sparsity and various low-rank tensor formats, as well as linear combinations thereof. When Gaussian random matrices are used for sketching, STTA is admissible to an analysis that builds and extends upon existing results on the generalized Nyström approximation for matrices. Our results show that STTA can be expected to attain a nearly optimal approximation error if the sizes of the sketches are suitably chosen. A range of numerical experiments illustrates the performance of STTA compared to existing deterministic and randomized approaches.
