Statistical methods relating tensor predictors to scalar outcomes in a regression model generally vectorize the tensor predictor and estimate the coefficients of its entries employing some form of regularization, use summaries of the tensor covariate, or use a low dimensional approximation of the coefficient tensor. However, low rank approximations of the coefficient tensor can suffer if the true rank is not small. We propose a tensor regression framework which assumes a soft version of the parallel factors (PARAFAC) approximation. In contrast to classic PARAFAC, where each entry of the coefficient tensor is the sum of products of row-specific contributions across the tensor modes, the soft tensor regression (Softer) framework allows the row-specific contributions to vary around an overall mean. We follow a Bayesian approach to inference, and show that softening the PARAFAC increases model flexibility, leads to improved estimation of coefficient tensors, more accurate identification of important predictor entries, and more precise predictions, even for a low approximation rank. From a theoretical perspective, we show that employing Softer leads to a weakly consistent posterior distribution of the coefficient tensor, irrespective of the true or approximation tensor rank, a result that is not true when employing the classic PARAFAC for tensor regression. In the context of our motivating application, we adapt Softer to symmetric and semi-symmetric tensor predictors and analyze the relationship between brain network characteristics and human traits.soft
翻译:在回归模型中,将电压预测器与卡路里结果联系起来的统计方法,一般地将电压预测器向上推,并估计其条目的系数,采用某种形式的正规化,使用高正同异变汇总,或使用系数高的低维近似值。然而,如果真实级别不小,系数高的低等级近似值可能会受到影响。我们提议了一个慢压回归框架,以软版本的平行因素近似值(PARAFAC)为假设值。与传统的PARAFFAC相比,系数拉值的每个条目是不同色调模式各行特定贡献产品的总和,软色调回归(Softer)框架允许特定行的贡献在整体平均值上有所变化。我们采用巴伊西亚的推论方法来推断,并表明软化PARAFAC模型增加了灵活性,从而改进了对重要预测器项(PARFAC)的预测值,更准确地确定重要的预测值,甚至更精确的预测值,即使是低近似等级。从理论角度看,我们发现,使用索福特尔的每组(Softer)导致不同行贡献的后后后,具体的里程里程(Sharimal)里程(Sharimal)里)里程关系是使用我们采用沙拉(Sharimal)的软化)的里程(Sharimalim)的里程)。