The Union of Manifolds Hypothesis

Deep learning has had tremendous success at learning low-dimensional representations of high-dimensional data. This success would be impossible if there was no hidden low-dimensional structure in data of interest; this existence is posited by the manifold hypothesis, which states that the data lies on an unknown manifold of low intrinsic dimension. In this paper, we argue that this hypothesis does not properly capture the low-dimensional structure typically present in image data. Assuming that data lies on a single manifold implies intrinsic dimension is identical across the entire data space, and does not allow for subregions of this space to have a different number of factors of variation. To address this deficiency, we put forth the union of manifolds hypothesis, which states that data lies on a disjoint union of manifolds of varying intrinsic dimensions. We empirically verify this hypothesis on commonly-used image datasets, finding that indeed, observed data lies on a disconnected set and that intrinsic dimension is not constant. We also provide insights into the implications the union of manifolds hypothesis has for deep learning, both supervised and unsupervised, showing that designing models with an inductive bias for this structure improves performance across classification and generative modelling tasks.