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 consider 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 of the union of manifolds hypothesis in 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. Our code is available at
this https URL.

本文讨论了流形假设不适当地捕捉了图像数据中典型的低维结构。我们考虑流形联合假设，并在常用图像数据集上进行实证验证，发现观察到的数据在一个不相连的集合上。我们还发现，针对这种结构设计具有归纳偏置的模型可以提高分类和生成建模任务的性能。