We introduce an algorithm to compute 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 ǫ, 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 in combination with the Alternating Least Squares (ALS) algorithm. We present examples of its use within a convergent iteration to compute inverse operators in high dimensions. We also briefly discuss the spectral norm as a computational alternative to the Frobenius norm in estimating approximation errors of tensor ID.We reduce the problem of finding tensor IDs to that of constructing Interpolative Decompositions of certain matrices. These matrices are generated via randomized projection of the terms of the given tensor. We provide cost estimates and several examples of the new approach to the reduction of separation rank.

We describe an algorithm for the reparametrization of a closed curve defined by a sequence of m points while providing the user a high level of control over the frequency content of the resulting curve. Specifically, the algorithm views the tangential angle of the curve as a function of the arc-length, filters it as such, and adds a small analytic perturbation so that the curve passes through the input data. If the number of nodes in the initial discretization is n, the entire scheme has asymptotic complexity O(n log n). The resulting curve is analytic and bandlimited. The performance of the scheme is illustrated with several numerical examples.

No abstract