  Considering smooth mappings from input vectors to continuous targets, our
goal is to characterise subspaces of the input domain, which are invariant
under such mappings. Thus, we want to characterise manifolds implicitly defined
by level sets. Specifically, this characterisation should be of a global
parametric form, which is especially useful for different informed data
exploration tasks, such as building grid-based approximations, sampling points
along the level curves, or finding trajectories on the manifold. However,
global parameterisations can only exist if the level sets are connected. For
this purpose, we introduce a novel and flexible class of neural networks that
generalise input-convex networks. These networks represent functions that are
guaranteed to have connected level sets forming smooth manifolds on the input
space. We further show that global parameterisations of these level sets can be
always found efficiently. Lastly, we demonstrate that our novel technique for
characterising invariances is a powerful generative data exploration tool in
real-world applications, such as computational chemistry.
