Hierarchically low-rank matrices, fast multipole methods, Kronecker factorization, decomposition, matrix-vector, linear alegbra
Exploiting structures of matrices goes beyond identifying their non-zero patterns. In many cases, dense full-rank matrices have low-rank submatrices that can be exploited to construct fast approximate algorithms. In other cases, dense matrices can be decomposed into Kronecker factors that are much smaller than the original matrix. Sparsity is a consequence of the connectivity of the underlying geometry (mesh, graph, interaction list, etc.), whereas the rank-deficiency of submatrices is closely related to the distance within this underlying geometry. For high dimensional geometry encountered in data science applications, the curse of dimensionality poses a challenge for rank-structured approaches. On the other hand, models in data science that are formulated as a composition of functions, lead to a Kronecker product structure that yields a different kind of fast algorithm. In this lecture, we will look at some examples of when rank structure and Kronecker structure can be useful.
Yokota, R. (2021). Lecture 09: Hierarchically Low Rank and Kronecker Methods. Mathematical Sciences Spring Lecture Series. Retrieved from https://scholarworks.uark.edu/mascsls/16