A natural question at the intersection of universality efforts and manifold learning is the following: What kinds of architecture are universal approximators of maps between manifolds that are topologically interesting? A (low-dimensional) manifold hypothesis has been underlying the study of inverse problems ensuring Lipschitz stability, implying a like-wise hypothesis for data. This is used, for example, in inference through flows. By exploiting the topological parallels between locally bilipschitz maps, covering spaces, and local homeomorphisms, we find that a novel network of the form p o E, where E is an injective flow and p a coordinate projection, is a universal approximator of local diffeomorphisms between compact smooth (sub)manifolds embedded in Euclidean spaces. We show that the network allows for the computation of multi-valued inversion and that our analysis holds in the interesting case when the target map between manifolds changes topology and its degree is a priori not known. We also show that the network can be used, for example, in supervised problems for recovering the group action of a group invariant map if the group is finite, and in unsupervised problems by informing the choice of topologically expressive starting spaces in the generative case.
De Eero Simoncelli
De Naoki Saito
De Maarten de Hoop
De Martin Vetterli
De Maureen Clerc
De Irène Waldspurger
De Jérôme Kalifa
De Gitta Kutyniok
De Michael Unser
De Edouard Oyallon
De Remi Gribonval
De Ronnie Pavlov
De Viviane Pons
De Wolfgang Arendt
De David Hewett
