← 研踪 PaperTrailarXiv:2610.34273 · v2
Asymptotic Rank of Weighted Graph Tensors via Completely Positive Covers
首次提交 2026-10-25 21:03
版本更新 2026-11-03 21:38
雷达观测 2026-11-04 23:39
摘要
We give a conic upper bound on the asymptotic rank of every weighted graph tensor over every field. If $W$ is the logarithmic edge-weight matrix of its dimension matrix $Q$, then $$ \operatorname{MaxCut}(W)\le\log_2\operatorname{\widetilde R}_{\mathbb F}(\operatorname{WT}(Q))\le U_{\operatorname{CP}}^Δ(W). $$ Here $U_{\operatorname{CP}}^Δ(W)$ maximizes $\sum_{1\le i<j\le n}W_{i,j}P_{i,j}$ over nonnegative symmetric zero-diagonal matrices $P$ such that $J-2P$ is copositive and $P_{i,j}\le P_{i,k}+P_{k,j}$ for all distinct vertices. The matrix $J$ is all-one, and copositivity means nonnegativity of the quadratic form on nonnegative vectors. Our starting point extends OpenAI's polynomial multiplication method and its subsequent arbitrary-field developments from three tensor legs to arbitrary order. We prove $$ \left\lfloor\frac{n^2}{4}\right\rfloor\le\log_2\operatorname{\widetilde R}_{\mathbb F}(\operatorname{GT}_2(K_n))\le\frac{n^2}{4}. $$ The edge exponent is at most $n/(2(n-1))$ and tends to the optimal value $1/2$, improving the previous Coppersmith--Winograd bounds and answering Question 1.3.1 of Christandl, Vrana, and Zuiddam. The bounds coincide for every even $n$. For every odd $m\ge3$, the theorem gives $\log_2\widetilde R_{\mathbb F}(\operatorname{GT}_2(C_m))\le m-1+1/(m+1)$, improving the previous cycle bound with a gap of only $1/(m+1)$ above the maximum cut. On weighted triangles, the program recovers the rectangular matrix multiplication curve of Uznański and Amanov, van den Berg, and Zuiddam. The CP and semidefinite programs, with or without path constraints, all have value $9/4$ on the unweighted triangle, giving this upper bound a natural conic interpretation. Further applications give explicit bounds for complete multipartite graphs, split graphs, and wheels, including new cases of maximum-cut equality.
作者备注：80 pages, 5 figures, abstract shortened to meet arXiv requirements
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v2 · 2026-11-03 21:38 · 最新观测版本
Asymptotic Rank of Weighted Graph Tensors via Completely Positive Covers
We give a conic upper bound on the asymptotic rank of every weighted graph tensor over every field. If $W$ is the logarithmic edge-weight matrix of its dimension matrix $Q$, then $$ \operatorname{MaxCut}(W)\le\log_2\operatorname{\widetilde R}_{\mathbb F}(\operatorname{WT}(Q))\le U_{\operatorname{CP}}^Δ(W). $$ Here $U_{\operatorname{CP}}^Δ(W)$ maximizes $\sum_{1\le i<j\le n}W_{i,j}P_{i,j}$ over nonnegative symmetric zero-diagonal matrices $P$ such that $J-2P$ is copositive and $P_{i,j}\le P_{i,k}+P_{k,j}$ for all distinct vertices. The matrix $J$ is all-one, and copositivity means nonnegativity of the quadratic form on nonnegative vectors. Our starting point extends OpenAI's polynomial multiplication method and its subsequent arbitrary-field developments from three tensor legs to arbitrary order. We prove $$ \left\lfloor\frac{n^2}{4}\right\rfloor\le\log_2\operatorname{\widetilde R}_{\mathbb F}(\operatorname{GT}_2(K_n))\le\frac{n^2}{4}. $$ The edge exponent is at most $n/(2(n-1))$ and tends to the optimal value $1/2$, improving the previous Coppersmith--Winograd bounds and answering Question 1.3.1 of Christandl, Vrana, and Zuiddam. The bounds coincide for every even $n$. For every odd $m\ge3$, the theorem gives $\log_2\widetilde R_{\mathbb F}(\operatorname{GT}_2(C_m))\le m-1+1/(m+1)$, improving the previous cycle bound with a gap of only $1/(m+1)$ above the maximum cut. On weighted triangles, the program recovers the rectangular matrix multiplication curve of Uznański and Amanov, van den Berg, and Zuiddam. The CP and semidefinite programs, with or without path constraints, all have value $9/4$ on the unweighted triangle, giving this upper bound a natural conic interpretation. Further applications give explicit bounds for complete multipartite graphs, split graphs, and wheels, including new cases of maximum-cut equality.
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