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Theory of Computing Systems

, Volume 58, Issue 4, pp 506–527 | Cite as

The Arithmetic Complexity of Tensor Contraction

  • Florent Capelli
  • Arnaud Durand
  • Stefan Mengel
Article
  • 151 Downloads

Abstract

We investigate the algebraic complexity of tensor calculus. We consider a generalization of iterated matrix product to tensors and show that the resulting formulas exactly capture V P, 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.

Keywords

Algebraic complexity Arithmetic circuits Tensor calculus 

Notes

Acknowledgments

We thank Yann Strozecki for a detailed and helpful feedback on an early version of this paper. We also thank Hervé Fournier, Guillaume Malod and Sylvain Perifel for helpful discussions.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.IMJ UMR 7586 - LogiqueUniversité Paris DiderotParisFrance
  2. 2.LIX UMR 7161, Ecole PolytechniquePalaiseauFrance

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