The arithmetic complexity of tensor contractions
  We investigate the algebraic complexity of tensor calulus. We consider a generalization of iterated matrix product to tensors and show that the resulting formulas exactly capture VP, the class of polynomial families efficiently computable by arithmetic circuits. This gives a natural and robust characterization of this complexity class that despite its naturalness is not very well understood so far. 
Florent Capelli, Arnaud Durand, Stefan Mengel
cs.CC