Fast sparse matrix multiplication

Let <i>A</i> and <i>B</i> two <i>n</i>×<i>n</i> matrices over a ring <i>R</i> (e.g., the reals or the integers) each containing at most <i>m</i> nonzero elements. We present a new algorithm that multiplies <i>A</i> and <i>B</i> using <i>O</i>(<i>m</i>^0.7<i>n</i>^1.2+<i>n</i>^{2+<i>o</i>(1)}) algebraic operations (i.e., multiplications, additions and subtractions) over <i>R</i>. The naïve matrix multiplication algorithm, on the other hand, may need to perform Ω(<i>mn</i>) operations to accomplish the same task. For <i>m</i>≤<i>n</i>^1.14, the new algorithm performs an almost optimal number of only <i>n</i>^{2+<i>o</i>(1)} operations. For <i>m</i>≤<i>n</i>^1.68, the new algorithm is also faster than the best known matrix multiplication algorithm for dense matrices which uses <i>O</i>(<i>n</i>^2.38) algebraic operations. The new algorithm is obtained using a surprisingly straightforward combination of a simple combinatorial idea and existing fast <i>rectangular</i> matrix multiplication algorithms. We also obtain improved algorithms for the multiplication of more than two sparse matrices. As the known fast rectangular matrix multiplication algorithms are far from being practical, our result, at least for now, is only of theoretical value.