Algebraic methods for tensor data

We develop algebraic methods for computations with tensor data. We give three applications: extracting features that are invariant under the orthogonal symmetries in each of the modes, approximation of the tensor spectral norm, and amplification of low rank tensor structure. We introduce colored Brauer diagrams, which are used for algebraic computations and in analyzing their computational complexity. We present numerical experiments whose results show that the performance of the alternating least squares algorithm for rank 1 approximations for tensors can be improved using tensor amplification.