  We consider $N$-way data arrays and low-rank tensor factorizations where the
time mode is coded as a sparse linear combination of temporal elements from an
over-complete library. Our method, Shape Constrained Tensor Decomposition
(SCTD) is based upon the CANDECOMP/PARAFAC (CP) decomposition which produces
$r$-rank approximations of data tensors via outer products of vectors in each
dimension of the data. By constraining the vector in the temporal dimension to
known analytic forms which are selected from a large set of candidate
functions, more readily interpretable decompositions are achieved and analytic
time dependencies discovered. The SCTD method circumvents traditional {\em
flattening} techniques where an $N$-way array is reshaped into a matrix in
order to perform a singular value decomposition. A clear advantage of the SCTD
algorithm is its ability to extract transient and intermittent phenomena which
is often difficult for SVD-based methods. We motivate the SCTD method using
several intuitively appealing results before applying it on a number of
high-dimensional, real-world data sets in order to illustrate the efficiency of
the algorithm in extracting interpretable spatio-temporal modes. With the rise
of data-driven discovery methods, the decomposition proposed provides a viable
technique for analyzing multitudes of data in a more comprehensible fashion.
