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When analyzing parametric statistical models, a useful approach consists inmodeling geometrically the parameter space. However, even for very simple andcommonly used hierarchical models like statistical mixtures or stochastic deepneural networks, the smoothness assumption of manifolds is violated at singularpoints which exhibit non-smooth neighborhoods in the parameter space. Thesesingular models have been analyzed in the context of learning dynamics, wheresingularities can act as attractors on the learning trajectory and, therefore,negatively influence the convergence speed of models. We propose a generalapproach to circumvent the problem arising from singularities by usingstratifolds, a concept from algebraic topology, to formally model singularparameter spaces. We use the property that specific stratifolds are equippedwith a resolution method to construct a smooth manifold approximation of thesingular space. We empirically show that using (natural) gradient descent onthe smooth manifold approximation instead of the singular space allows us toavoid the attractor behavior and therefore improve the convergence speed inlearning.

Towards Modeling and Resolving Singular Parameter Spaces using Stratifolds