  In this paper, we propose Total Variation Regularized Tensor-on-scalar
Regression(TVTR), a novel method for estimating the association between a
tensor outcome (a one dimensional or multidimensional array) and scalar
predictors. While the statistical developments proposed here were motivated by
the brain mapping and activity tracking, the methodology is designed and
presented in generality and is applicable to many other areas of scientific
research. The estimator is the solution of a penalized regression problem where
the objective is the sum of square error plus a total variation (TV)
regularization on the predicted mean across all subjects. We propose an
algorithm for the parameter estimation, which is efficient and scalable in
distributed computing platform. Proof of the algorithm convergence is provided
and the statistical consistency of the estimator is presented via an oracle
inequality. We presented 1D and 2D simulation results, and demonstrate that
TVTR outperforms existing methods in most cases. We also demonstrate the
general applicability of the method by two real data examples including the
analysis of the 1D accelerometry subsample of a large community-based study for
mood disorders and the analysis of the 3D MRI data from the attention
deficient/hyperactive deficient (ADHD) 200 consortium.
