This isn't quite right. Moment methods (that rely on tensor decompositions) have a few problems:

(i) they have convergence bounds, but in practice need more data than we have available

(ii) they don't do as well as EM usually, but using them to initialize parameters for EM sometimes does better than EM with random initialization schemes

(iii) it turns out variational methods can also be embarrassingly parallelized without losing much accuracy in practice

(iv) right now moment methods don't work for arbitrary graphical models