Common idea in Karatsuba, Gauss and Strassen multiplication

Question

The identities used in multiplication algorithms by

• Karatsuba (integers)

• Gauss (complex numbers)

• Strassen (matrices)

seem very closely related. Is there a common abstract framework/generalization?

Answer 1

A partial answer to your question is the group-theoretic approach first developed by Cohn and Umans and further developed by Cohn, Kleinberg, Szegedy and Umans. It can "sort of" capture Strassen and Coppersmith-Winograd for matrix multiplication.

Answer 2

The classical framework is the one of bilinear algorithms and tensor rank decompositions; basically, you construct the 3-way tensor associated to the bilinear map $f(A,B) = A \cdot B$, in the basis of the coefficients, then look for a decomposition of it as a sum of rank-one tensors (i.e., those of the form $T_{i,j,k} = u_i v_j w_k$). You'll find this explained in more detail, for instance, in this article by Bläser, or in the book by Bürgisser, Clausen, Shokrollahi, Algebraic Complexity Theory.

As far as I understand, the reformulation in terms of group respresentations that Suresh mentions in his answer is a later one, and I find it less suitable for a first approach to the subject (but, of course, that might be bias on my part).