  We introduce tensor Interpolative Decomposition (tensor ID) for the reduction
of the separation rank of Canonical Tensor Decompositions (CTDs). Tensor ID
selects, for a user-defined accuracy \epsilon, a near optimal subset of terms
of a CTD to represent the remaining terms via a linear combination of the
selected terms. Tensor ID can be used as an alternative to or a step of the
Alternating Least Squares (ALS) algorithm. In addition, we briefly discuss
Q-factorization to reduce the size of components within an ALS iteration.
Combined, tensor ID and Q-factorization lead to a new paradigm for the
reduction of the separation rank of CTDs. In this context, we also discuss the
spectral norm as a computational alternative to the Frobenius norm.
  We reduce the problem of finding tensor IDs to that of constructing
Interpolative Decompositions of certain matrices. These matrices are generated
via either randomized projection or randomized sampling of the given tensor. We
provide cost estimates and several examples of the new approach to the
reduction of separation rank.
