In this note, we briefly present a generalized tensor CUR (GTCUR) approximation for tensor pairs (X,Y) and tensor triplets (X,Y,Z) based on the tubal product (t-product). We use the tensor Discrete Empirical Interpolation Method (TDEIM) to do these extensions. We show how the TDEIM can be utilized to generalize the classical tensor CUR (TCUR) approximation, which acts only on a single tensor, to jointly compute the TCUR of two and three tensors. This approach can be used to sample relevant lateral/horizontal slices of one data tensor relative to one or two other data tensors. For some special cases, the Generalized TCUR (GTCUR) approximation is reduced to the classical TCUR for both tensor pairs and tensor triplets in a similar fashion as shown for the matrices.

翻译：本注记简要介绍了基于管积（t-product）的张量对（X,Y）与张量三元组（X,Y,Z）的广义张量CUR（GTCUR）逼近方法。我们利用张量离散经验插值方法（TDEIM）实现上述扩展，并展示如何通过TDEIM将仅作用于单个张量的经典张量CUR（TCUR）逼近推广为两个及三个张量的联合TCUR计算。该方法能够实现对某个数据张量相对于一个或两个其他数据张量的相关侧向/水平切片进行采样。在特定情形下，广义TCUR（GTCUR）逼近在张量对与张量三元组中均能退化为经典TCUR，其退化方式与矩阵情况类似。