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- W2969598375 abstract "We show that two important quantities from two disparate areas of complexity theory --- Strassen's exponent of matrix multiplication $omega$ and Grothendieck's constant $K_G$ --- are intimately related. They are different measures of size for the same underlying object --- the matrix multiplication tensor, i.e., the $3$-tensor or bilinear operator $mu_{l,m,n} : mathbb{F}^{l times m} times mathbb{F}^{m times n} to mathbb{F}^{l times n}$, $(A,B) mapsto AB$ defined by matrix-matrix product over $mathbb{F} = mathbb{R}$ or $mathbb{C}$. It is well-known that Strassen's exponent of matrix multiplication is the greatest lower bound on (the log of) a tensor rank of $mu_{l,m,n}$. We will show that Grothendieck's constant is the least upper bound on a tensor norm of $mu_{l,m,n}$, taken over all $l, m, n in mathbb{N}$. Aside from relating the two celebrated quantities, this insight allows us to rewrite Grothendieck's inequality as a norm inequality [ lVertmu_{l,m,n}rVert_{1,2,infty} =max_{X,Y,Mneq0}frac{|operatorname{tr}(XMY)|}{lVert XrVert_{1,2}lVert YrVert_{2,infty}lVert MrVert_{infty,1}}le K_G. ] We prove that Grothendieck's inequality is unique: If we generalize the $(1,2,infty)$-norm to arbitrary $p,q, r in [1, infty]$, [ lVertmu_{l,m,n}rVert_{p,q,r}=max_{X,Y,Mneq0}frac{|operatorname{tr}(XMY)|}{|X|_{p,q}|Y|_{q,r}|M|_{r,p}}, ] then $(p,q,r )=(1,2,infty)$ is, up to cyclic permutations, the only choice for which $lVertmu_{l,m,n}rVert_{p,q,r}$ is uniformly bounded by a constant independent of $l,m,n$." @default.
- W2969598375 created "2019-08-29" @default.
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- W2969598375 date "2019-08-22" @default.
- W2969598375 modified "2023-10-18" @default.
- W2969598375 title "Grothendieck constant is norm of Strassen matrix multiplication tensor" @default.
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- W2969598375 doi "https://doi.org/10.1007/s00211-019-01070-6" @default.
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