Bad and Good News for Strassen’s Laser Method: Border Rank of $$\mathrm{Perm}_3$$ and Strict Submultiplicativity
Bad and Good News for Strassen’s Laser Method: Border Rank of $$\mathrm{Perm}_3$$ and Strict Submultiplicativity
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Strassen 激光方法的坏消息和好消息：$$mathrm{Perm}_3$$ 的边界等级和严格次乘性
DOI：
10.1007/s10208-022-09579-3
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发表时间：
2023
影响因子：
3
通讯作者：
Landsberg, J. M.
中科院分区：
文献类型：
--
作者：
Conner, Austin;Huang, Hang;Landsberg, J. M.
我们确定张量的边界等级，这些张量可能会提前已知矩阵乘法指数的上限。小铜史密斯-威诺格拉德张量的克罗内克平方等于永久张量，并且有可能用于显示。我们证明了复杂性理论的负面结果，即其边界等级为 16，解决了一个长期存在的问题。关于它的偏斜表亲 in（可能用于证明），我们显示其克罗内克平方的边界秩最多为 42，这是一个显着的次乘性结果，因为其边界秩的平方为 64。我们还确定了小 Coppersmith-Winograd 张量的模空间 VSP。
We determine the border ranks of tensors that could potentially advance the known upper bound for the exponentof matrix multiplication. The Kronecker square of the smallCoppersmith–Winograd tensor equals thepermanent, and could potentially be used to show. We prove the negative result for complexity theory that its border rank is 16, resolving a longstanding problem. Regarding itsskew cousin in, which could potentially be used to prove, we show the border rank of its Kronecker square is at most 42, a remarkable sub-multiplicativity result, as the square of its border rank is 64. We also determine moduli spaces VSP for the small Coppersmith–Winograd tensors.