Bundle Networks: Fiber Bundles, Local Trivializations, and a Generative Approach to Exploring Many-to-one Maps
Keywords: generative models, applications of topology to deep learning, many-to-one maps, invertible neural nets
Abstract: Many-to-one maps are ubiquitous in machine learning, from the image recognition model that assigns a multitude of distinct images to the concept of “cat” to the time series forecasting model which assigns a range of distinct time-series to a single scalar regression value. While the primary use of such models is naturally to associate correct output to each input, in many problems it is also useful to be able to explore, understand, and sample from a model's fibers, which are the set of input values $x$ such that $f(x) = y,$ for fixed $y$ in the output space. In this paper we show that popular generative architectures are ill-suited to such tasks. Motivated by this, we introduce a novel generative architecture, Bundle Networks, based on the concept of a fiber bundle from (differential) topology. BundleNets exploit the idea of a local trivialization wherein a space can be locally decomposed into a product space that cleanly encodes the many-to-one nature of the map. By enforcing this decomposition in BundleNets and by utilizing state-of-the-art invertible components, investigating a network's fibers becomes natural.
One-sentence Summary: We draw from the theory of fiber bundles in (differential) topology to create a principled approach to generative models that allow us to learn and sample from the "fiber" over a point in a many-to-one map.