Abstract The emergence of large-scale data demands new regression models with multi-dimensional coefficient arrays, known as tensor regression models. The recently proposed tensor ring decomposition has interesting properties of enhanced representation and compression capability, cyclic permutation invariance and balanced tensor ring rank, which may lead to efficient computation and fewer parameters in regression problems. In this paper, a generally multi-linear tensor-on-tensor regression model is proposed that the coefficient array has a low-rank tensor ring structure, which is termed tensor ring ridge regression (TRRR). Two optimization models are developed for the TRRR problem and solved by different algorithms: the tensor factorization based one is solved by alternating least squares algorithm, and accelerated by a fast network contraction, while the rank minimization based one is addressed by the alternating direction method of multipliers algorithm. Comparative experiments, including Spatio-temporal forecasting tasks and 3D reconstruction of human motion capture data from its temporally synchronized video sequences, demonstrate the enhanced performance of our algorithms over existing state-of-the-art ones, especially in terms of training time.

中文翻译：

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多线性回归的低秩张量环学习

摘要 大规模数据的出现需要具有多维系数数组的新回归模型，称为张量回归模型。最近提出的张量环分解具有增强表示和压缩能力、循环置换不变性和平衡张量环秩的有趣特性，这可能导致回归问题中的高效计算和更少的参数。本文提出了一种广义多线性张量对张量回归模型，其系数阵列具有低秩张量环结构，称为张量环脊回归（TRRR）。针对 TRRR 问题开发了两种优化模型并通过不同的算法求解：基于张量分解的一种通过交替最小二乘算法求解，并通过快速网络收缩加速，而基于秩最小化的方法是通过乘法器算法的交替方向方法来解决的。比较实验，包括时空预测任务和来自其时间同步视频序列的人体运动捕捉数据的 3D 重建，证明了我们的算法优于现有最先进算法的性能，尤其是在训练时间方面。