Peter released a technical report (available from arxiv) on the computation of the Tucker decomposition on large tensors: the Tucker decomposition is just a multidimensional generalization of the Singular Value Decomposition (SVD). The report includes a new algorithm designed by Peter which is more accurate than competing Matlab implementations, in the case where you have very large tensors (3 or 4 dimensional) and need external memory computations.
There exist incremental SVD algorithms. It does seem to me that a nice property of Turney’s tensor algorithm is that it can be made part of an incremental scheme efficiently.
Another challenge would be to have a serious look at parallel implementations. I think that Turney’s scheme could benefit tremendously from several processors.