The Arithmetic Complexity of Tensor Contraction

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.

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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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Correspondence to Florent Capelli.

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Partially supported by ANR Blanc CompA ANR-13-BS02-0001

Partially supported by a grant from Quallcom.

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Capelli, F., Durand, A. & Mengel, S. The Arithmetic Complexity of Tensor Contraction. Theory Comput Syst 58, 506–527 (2016). https://doi.org/10.1007/s00224-015-9630-8

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Keywords

  • Algebraic complexity
  • Arithmetic circuits
  • Tensor calculus