Researchers have demonstrated that low-dimensional training of deep neural networks can yield high accuracy. However, the mechanisms whereby spanning pruning, lottery tickets, and training within random subspaces (among other approaches to sparsification) work is not well understood. In this paper, Larsen et al. apply Gordon’s escape theorem (a tool from high-dimensional probability theory) to provide a theoretical explanation for the existence of a threshold training dimension for a specified model and task. In addition, they introduce a new scientific method – lottery subspaces – which use even fewer dimensions than other approaches to train deep nets.