Sparse Matrices Powering Three Pillars of Science: Simulation, Data, and Learning

Abstract

In addition to the traditional theory and experimental pillars of science, we are witnessing the emergence of three more recent pillars, which are simulation, data analysis, and machine learning. All three recent pillars of science rely on computing but in different ways. Matrices, and sparse matrices in particular, play an outsized role in all three computing related pillars of science, which will be the topic of my talk. Solving systems of linear equations have traditionally driven research in sparse matrix computation for decades. Direct and iterative solvers, together with finite element computations, still account for the primary use case for sparse matrix data structures and algorithms. These solvers are the workhorses of scientific simulations. Modern methods for data analysis, such as matrix decompositions and graph analytics, often use the same underlying sparse matrix technology. The same can be said for various machine learning methods, where the data and/or the models are often sparse. I highlight some of the emerging use cases of sparse matrices outside the domain of solvers. These include graph computations, computational biology and emerging techniques in machine learning. A recurring theme in all these novel use cases is the concept of a semiring on which the sparse matrix computations are carried out. By overloading scalar addition and multiplication operators of a semiring, we can attack a much richer set of computational problems using the same sparse data structures and algorithms. This approach has been formalized by the GraphBLAS effort. I will illustrate one example application from each problem domain, together with the most computationally demanding sparse matrix primitive required for its efficient execution. I will also cover available software, such as various implementations of the GraphBLAS standard, that implement these sparse matrix primitives efficiently on various architectures.