  We develop algebraic methods for computations with tensor data. We give 3
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 square algorithm for the low rank
approximation of tensors can be improved using tensor amplification.
