An \~{O}(n2)(n^2) Time Matrix Multiplication Algorithm

We show, for the input vectors $(a_0, a_1, ..., a_{n-1})$ and $(b_0, b_1, ..., b_{n-1})$, where $a_i$'s and $b_j$'s are real numbers, after \~{O}$(n)$ time preprocessing for each of them, the vector multiplication $(a_0, a_1, ..., a_{n-1})(b_0, b_1, ..., b_{n-1})^T$ can be computed in \~{O}$(1)$ time. This enables the matrix multiplication of two $n\times n$ matrices to be computed in \~{O}$(n^2)$ time.Comment: Version 11 and Version 12 section 2 laid the foundation of this algorithm but has a problem unresolved. This version corrects the problem in Version 11 and Section 2 of Version 1

Stochastic Block Coordinate Frank-Wolfe Algorithm for Large-Scale Biological Network Alignment

With increasingly "big" data available in biomedical research, deriving accurate and reproducible biology knowledge from such big data imposes enormous computational challenges. In this paper, motivated by recently developed stochastic block coordinate algorithms, we propose a highly scalable randomized block coordinate Frank-Wolfe algorithm for convex optimization with general compact convex constraints, which has diverse applications in analyzing biomedical data for better understanding cellular and disease mechanisms. We focus on implementing the derived stochastic block coordinate algorithm to align protein-protein interaction networks for identifying conserved functional pathways based on the IsoRank framework. Our derived stochastic block coordinate Frank-Wolfe (SBCFW) algorithm has the convergence guarantee and naturally leads to the decreased computational cost (time and space) for each iteration. Our experiments for querying conserved functional protein complexes in yeast networks confirm the effectiveness of this technique for analyzing large-scale biological networks