Tensor decompositions such as the canonical format and the tensor train format have been widely utilized to reduce storage costs and operational complexities for high‐dimensional data, achieving linear scaling with the input dimension instead of exponential scaling. In this paper, we investigate even lower storage‐cost representations in the tensor ring format, which is an extension of the tensor train format with variable end‐ranks. Firstly, we introduce two algorithms for converting a tensor in full format to tensor ring format with low storage cost. Secondly, we detail a rounding operation for tensor rings and show how this requires new definitions of common linear algebra operations in the format to obtain storage‐cost savings. Lastly, we introduce algorithms for transforming the graph structure of graph‐based tensor formats, with orders of magnitude lower complexity than existing literature. The efficiency of all algorithms is demonstrated on a number of numerical examples, and in certain cases, we demonstrate significantly higher compression ratios when compared to previous approaches to using the tensor ring format.

中文翻译：

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关于张量环分解的算法和计算

张量分解（例如规范格式和张量列格式）已被广泛用于降低高维数据的存储成本和操作复杂性，从而实现了输入维线性缩放而不是指数缩放。在本文中，我们以张量环格式研究了甚至更低的存储成本表示形式，这是张量列格式具有可变最终秩的扩展。首先，我们介绍了两种算法，可将完整格式的张量转换为张量环格式，且存储成本低。其次，我们详细介绍了张量环的舍入运算，并说明了它如何要求以格式来通用线性代数运算的新定义，以节省存储成本。最后，我们介绍了用于转换基于图的张量格式的图结构的算法，与现有文献相比，其复杂度降低了几个数量级。在许多数值示例中证明了所有算法的效率，并且在某些情况下，与使用张量环格式的先前方法相比，我们证明了明显更高的压缩率。