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On the Algebraic Proof Complexity of Tensor Isomorphism

Authors Nicola Galesi, Joshua A. Grochow , Toniann Pitassi, Adrian She

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Nicola Galesi
  • Dipartimento Ingegneria Informatica Automatica e Gestionale "A. Ruberti", Sapienza University of Rome, Italy
Joshua A. Grochow
  • Departments of Computer Science and Mathematics, University of Colorado Boulder, CO, USA
Toniann Pitassi
  • Department of Computer Science, Columbia University, New York, NY, USA
Adrian She
  • Department of Mathematics and Computer Science, University of Toronto, Canada


NG and JAG would like to thank Michael Forbes for early conversation about the PC degree of matrix rank, which occurred at Dagstuhl Seminar 18051: Proof Complexity in early 2018. We would also like to thank the organizers A. Atserias, J. Nordstrom, P. Pudlák, and R. Santhanam for their invitation and support.

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Nicola Galesi, Joshua A. Grochow, Toniann Pitassi, and Adrian She. On the Algebraic Proof Complexity of Tensor Isomorphism. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 4:1-4:40, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


The Tensor Isomorphism problem (TI) has recently emerged as having connections to multiple areas of research within complexity and beyond, but the current best upper bound is essentially the brute force algorithm. Being an algebraic problem, TI (or rather, proving that two tensors are non-isomorphic) lends itself very naturally to algebraic and semi-algebraic proof systems, such as the Polynomial Calculus (PC) and Sum of Squares (SoS). For its combinatorial cousin Graph Isomorphism, essentially optimal lower bounds are known for approaches based on PC and SoS (Berkholz & Grohe, SODA '17). Our main results are an Ω(n) lower bound on PC degree or SoS degree for Tensor Isomorphism, and a nontrivial upper bound for testing isomorphism of tensors of bounded rank. We also show that PC cannot perform basic linear algebra in sub-linear degree, such as comparing the rank of two matrices (which is essentially the same as 2-TI), or deriving BA = I from AB = I. As linear algebra is a key tool for understanding tensors, we introduce a strictly stronger proof system, PC+Inv, which allows as derivation rules all substitution instances of the implication AB = I → BA = I. We conjecture that even PC+Inv cannot solve TI in polynomial time either, but leave open getting lower bounds on PC+Inv for any system of equations, let alone those for TI. We also highlight many other open questions about proof complexity approaches to TI.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • Theory of computation → Problems, reductions and completeness
  • Algebraic proof complexity
  • Tensor Isomorphism
  • Graph Isomorphism
  • Polynomial Calculus
  • Sum-of-Squares
  • reductions
  • lower bounds
  • proof complexity of linear algebra


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