  In this work, we study several variants of matrix reduction via Gaussian
elimination that try to keep the reduced matrix sparse. The motivation comes
from the growing field of topological data analysis where matrix reduction is
the major subroutine to compute barcodes, the main invariant therein. We
propose two novel variants of the standard algorithm, called swap and
retrospective reductions. We test them on a large collection of data against
other known variants to compare their efficiency, and we find that sometimes
they provide a considerable speed-up. We also present novel output-sensitive
bounds for the retrospective variant which better explain the discrepancy
between the cubic worst-case complexity bound and the almost linear practical
behavior of matrix reduction. Finally, we provide several constructions on
which one of the variants performs strictly better than the others.
