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 $R$ is standard Gaussian. For positively homogeneous nonpolynomial activations, which include sign, Heaviside, ReLU, and ReLU powers among others, we prove that $$n \gtrsim \frac{1}θ\max\left\{m,\log\left(\frac{1}δ\right)\right\}$$ neurons suffice for $σ(XR)$ to have full row rank $m$ with probability at least $1-δ$. This dimension-free bound exponentially improves the previous general-dimension...
| Authors | Luca Becchetti, Matteo Russo, Ruben Skorupinski |
|---|---|
| Primary category | cs.LG |
| Published | 30 September 2026 |
| Last updated at source | 30 September 2026 |
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