发表机构
Caltech（加州理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出在Lovitz-Petrov条件下三阶张量分解的多项式时间算法，结合多项式时间验证实现端到端唯一性认证，确保分解具有最小秩。
AI 中文摘要
可辨识性准则用于证明给定的张量分解是唯一的秩分解。Kruskal经典条件是已知最著名的确定性可辨识性准则之一。然而，在Kruskal条件下，目前没有已知的多项式时间分解算法，且验证该条件本身是NP难的。Lovitz和Petrov引入了一个严格更一般的可辨识性条件，该条件相比之下可以在多项式时间内验证，但此前在该条件下也没有已知的多项式时间分解算法。我们给出了在Lovitz-Petrov条件下三阶张量分解的多项式时间算法。此外，将我们的算法与Lovitz-Petrov条件的多项式时间验证相结合，产生了一个高效的端到端认证流程：在计算分解后，可以在多项式时间内确定性地证明该分解是唯一的，因此具有最小秩。这与任意张量分解形成对比，后者仅能证明张量秩的上界，而确定张量秩在一般情况下是NP难的。
英文摘要
Identifiability criteria certify that a given tensor decomposition is a unique rank decomposition. Kruskal's classical condition is one of the best-known deterministic criteria for identifiability. However, no polynomial-time decomposition algorithm is known under the Kruskal condition, and verifying the condition itself is NP-hard. Lovitz and Petrov introduced a strictly more general identifiability condition which, in contrast, is polynomial-time verifiable, but no polynomial-time decomposition algorithm was previously known under this condition. We give a polynomial-time algorithm for tensor decomposition under the Lovitz--Petrov condition. Moreover, combining our algorithm with polynomial-time verification of the Lovitz--Petrov condition yields an efficient end-to-end certification procedure: after computing a decomposition, one can deterministically certify in polynomial time that it is unique and therefore of minimum rank. This contrasts with an arbitrary tensor decomposition, which certifies only an upper bound on the tensor rank, while determining tensor rank is NP-hard in general.
CommentsGeneralized the decomposition algorithm from three-way tensors to m-way tensors
