  Optimization, a key tool in machine learning and statistics, relies on
regularization to reduce overfitting. Traditional regularization methods
control a norm of the solution to ensure its smoothness. Recently, topological
methods have emerged as a way to provide a more precise and expressive control
over the solution, relying on persistent homology to quantify and reduce its
roughness. All such existing techniques back-propagate gradients through the
persistence diagram, which is a summary of the topological features of a
function. Their downside is that they provide information only at the critical
points of the function. We propose a method that instead builds on
persistence-sensitive simplification and translates the required changes to the
persistence diagram into changes on large subsets of the domain, including both
critical and regular points. This approach enables a faster and more precise
topological regularization, the benefits of which we illustrate with
experimental evidence.
