Hierarchical Orthogonal Factorization: Sparse Least Squares Problems

In this work, we develop a fast hierarchical solver for solving large, sparse least squares problems. We build upon the algorithm, spaQR (sparsified QR), that was developed by the authors to solve large sparse linear systems. Our algorithm is built on top of a Nested Dissection based multifrontal QR approach. We use low-rank approximations on the frontal matrices to sparsify the vertex separators at every level in the elimination tree. Using a two-step sparsification scheme, we reduce the number of columns and maintain the ratio of rows to columns in each front without introducing any additional fill-in. With this improvised scheme, we show that the runtime of the algorithm scales as $\mathcal{O}(M \log N)$ and uses $\mathcal{O}(M)$ memory to store the factorization. This is achieved at the expense of a small and controllable approximation error. The end result is an approximate factorization of the matrix stored as a sequence of sparse orthogonal and upper-triangular factors and hence easy to apply/solve with a vector. Finally, we compare the performance of the spaQR algorithm in solving sparse least squares problems with direct multifrontal QR and CGLS iterative method with a standard diagonal preconditioner.