Grothendieck constant is norm of Strassen matrix multiplication tensor
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We show that two important quantities from two disparate areas of complexity theory --- Strassen's exponent of matrix multiplication ω 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 μ_l,m,n : F^l × m×F^m × n→F^l × n, (A,B) AB defined by matrix-matrix product over F = R or 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 μ_l,m,n. We will show that Grothendieck's constant is the least upper bound on a tensor norm of μ_l,m,n, taken over all l, m, n ∈N. Aside from relating the two celebrated quantities, this insight allows us to rewrite Grothendieck's inequality as a norm inequality μ_l,m,n_1,2,∞ =_X,Y,M≠0|tr(XMY)|/ X_1,2 Y_2,∞ M_∞,1≤ K_G, and thereby allows a natural generalization to arbitrary p,q, r, 1< p,q,r<∞. We show that the following generalization is locally sharp: μ_l,m,n_p,q,r=_X,Y,M≠0|tr(XMY)|/X_p,qY_q,rM_r,p≤ K_G · l^|1/q-1/2|· m^1-1/p· n^1/r, and conjecture that Grothendieck's inequality is unique: (p,q,r )=(1,2,∞) is the only choice for which μ_l,m,n_p,q,r is uniformly bounded by a constant. We establish two-thirds of this conjecture: uniform boundedness of μ_l,m,n_p,q,r over all l,m,n necessarily implies that {p,q,r}=1 and {p,q,r}=∞.
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