  Cycle representatives of persistent homology classes can be used to provide
descriptions of topological features in data. However, the non-uniqueness of
these representatives creates ambiguity and can lead to many different
interpretations of the same set of classes. One approach to solving this
problem is to optimize the choice of representative against some measure that
is meaningful in the context of the data. In this work, we provide a study of
the effectiveness and computational cost of several $\ell_1$-minimization
optimization procedures for constructing homological cycle bases for persistent
homology with rational coefficients in dimension one, including
uniform-weighted and length-weighted edge-loss algorithms as well as
uniform-weighted and area-weighted triangle-loss algorithms. We conduct these
optimizations via standard linear programming methods, applying general-purpose
solvers to optimize over column bases of simplicial boundary matrices.
  Our key findings are: (i) optimization is effective in reducing the size of
cycle representatives, (ii) the computational cost of optimizing a basis of
cycle representatives exceeds the cost of computing such a basis in most data
sets we consider, (iii) the choice of linear solvers matters a lot to the
computation time of optimizing cycles, (iv) the computation time of solving an
integer program is not significantly longer than the computation time of
solving a linear program for most of the cycle representatives, using the
Gurobi linear solver, (v) strikingly, whether requiring integer solutions or
not, we almost always obtain a solution with the same cost and almost all
solutions found have entries in {-1, 0, 1} and therefore, are also solutions to
a restricted $\ell_0$ optimization problem, and (vi) we obtain qualitatively
different results for generators in Erd\H{o}s-R\'enyi random clique complexes.
