The Structural Complexity of Matrix-Vector Multiplication

We consider the problem of preprocessing an n n matrix M, and supporting queries that, for any vector v, returns the matrix-vector product Mv. This problem has been extensively studied in both theory and practice: on one side, practitioners have developed algorithms that are highly efficient in practice, whereas on the other side, theoreticians have proven that the problem cannot be solved faster than naive multiplication in the worst-case. This lower bound holds even in the average-case, implying that existing average-case analyses cannot explain this gap between theory and practice. Hence, we study the problem for structured matrices. We show that for n n Boolean matrices of VC-dimension d, the matrix-vector multiplication problem can be solved with eO(n2)preprocessing and eO(n2 1/d) query time.