Partial and weighted matrix multiplication

In a paper published in 1981, Sch\"onhage showed that large total matrix multiplications can be reduced to powers of partial matrix multiplication tensors, which correspond to the bilinear computation task of multiplying matrices with some of the entries fixed to be zero. It was left as an open problem to generalize the method to the case when the multiplication is also partial in the sense that only a subset of the entries need to be computed. We prove a variant of a more general case: reducing large weighted matrix multiplications to tensor powers of a partial matrix multiplication in the sense that every entry of the result is a partial version of the inner product of the corresponding row and column of the factors that would appear in the usual matrix product. The implication is that support rank upper bounds on partial matrix multiplication tensors in this general sense give upper bounds on the support rank exponent of matrix multiplication.