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 $\Omega(n)$ 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, 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 $AB=I \rightarrow BA=I$. 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.

翻译：张量同构问题（TI）近年来崭露头角，与复杂性理论及更广泛领域的多个研究方向存在联系，但目前已知的最佳上界本质上仍是暴力搜索算法。作为一个代数问题，TI（更确切地说是证明两个张量非同构）天然适用于代数和半代数证明系统，例如多项式演算（PC）和平方和（SoS）方法。对于其组合学上的近亲——图同构问题，基于PC和SoS的方法已被证明具有本质上最优的下界（Berkholz & Grohe, SODA '17）。我们的主要结果包括：针对张量同构问题，给出PC度或SoS度的$\Omega(n)$下界；以及针对有界秩张量同构测试的非平凡上界。我们还证明了PC无法在次线性度数下完成基础线性代数运算，例如比较两个矩阵的秩，或从$AB=I$推导出$BA=I$。由于线性代数是理解张量的关键工具，我们引入了一个严格更强的证明系统PC+Inv，该系统允许将蕴含式$AB=I \rightarrow BA=I$的所有代入实例作为推导规则。我们推测即使PC+Inv也无法在多项式时间内求解TI，但未对任何方程组（更不用说TI对应的方程组）给出PC+Inv的下界。此外，我们还强调了关于TI证明复杂度方法的诸多其他未解决问题。