Dimension-Free Rank Lifting from Random Hyperplane Arrangements

We study the width required for a randomly initialized hidden layer of a neural network to achieve rank lifting. Namely, given a dataset $X \in \mathbb{R}^{m \times d}$ of $m$, $d$-dimensional input vectors separated by an angle of at least $θ$, we consider the random feature matrix $σ(XR)$, where $