  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
