  Dense prediction tasks such as depth perception and semantic segmentation are
important applications in computer vision that have a concrete topological
description in terms of partitioning an image into connected components or
estimating a function with a small number of local extrema corresponding to
objects in the image. We develop a form of topological regularization based on
persistent homology that can be used in dense prediction tasks with these
topological descriptions. Experimental results show that the output topology
can also appear in the internal activations of trained neural networks which
allows for a novel use of topological regularization to the internal states of
neural networks during training, reducing the computational cost of the
regularization. We demonstrate that this topological regularization of internal
activations leads to improved convergence and test benchmarks on several
problems and architectures.
