Many applications of regression study the predictors with complex forms such as tensors. Besides low dimensional assumption, the effects of predictors with a tensor structure typically show clustered or smooth phenomena in the sense that adjacent elements have a similar effect on the response. To simultaneously incorporate the low‐rank and smoothness in tensor regression, we generalize the CANDECOMP/PARAFAC (CP) decomposition to a smooth version and propose a novel regression model based on the smoothed decomposition. The asymptotic theory of the proposed method is studied, which shows a faster rate of convergence than the one without incorporating the smoothness. The experiments on both synthetic and real data confirm that the proposed method has advantages over the alternative especially when the sample size is small.

中文翻译：

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通过位置平滑改进的张量回归模型

回归的许多应用研究具有复杂形式（例如张量）的预测变量。除了低维假设外，具有张量结构的预测变量的影响通常还表现出聚簇或平滑现象，即相邻元素对响应的影响类似。为了在张量回归中同时包含低秩和平滑度，我们将CANDECOMP / PARAFAC（CP）分解推广到平滑版本，并基于平滑分解提出了一种新颖的回归模型。研究了该方法的渐近理论，该理论显示出比没有考虑平滑度的算法更快的收敛速度。通过对合成数据和真实数据的实验，证实了所提出的方法比其他方法更具优势，尤其是在样本量较小的情况下。