  Topological loss based on persistent homology has shown promise in various
applications. A topological loss enforces the model to achieve certain desired
topological property. Despite its empirical success, less is known about the
optimization behavior of the loss. In fact, the topological loss involves
combinatorial configurations that may oscillate during optimization. In this
paper, we introduce a general purpose regularized topology-aware loss. We
propose a novel regularization term and also modify existing topological loss.
These contributions lead to a new loss function that not only enforces the
model to have desired topological behavior, but also achieves satisfying
convergence behavior. Our main theoretical result guarantees that the loss can
be optimized efficiently, under mild assumptions.
