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Many-to-one maps are ubiquitous in machine learning, from the imagerecognition model that assigns a multitude of distinct images to the concept of"cat" to the time series forecasting model which assigns a range of distincttime-series to a single scalar regression value. While the primary use of suchmodels is naturally to associate correct output to each input, in many problemsit is also useful to be able to explore, understand, and sample from a model'sfibers, 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 generativearchitectures are ill-suited to such tasks. Motivated by this we introduce anovel generative architecture, a Bundle Network, based on the concept of afiber bundle from (differential) topology. BundleNets exploit the idea of alocal trivialization wherein a space can be locally decomposed into a productspace that cleanly encodes the many-to-one nature of the map. By enforcing thisdecomposition in BundleNets and by utilizing state-of-the-art invertiblecomponents, investigating a network's fibers becomes natural.

Bundle Networks: Fiber Bundles, Local Trivializations, and a Generative Approach to Exploring Many-to-one Maps