Let $X$ be a random $n \times d$ with independent whose distributions have densities (w.r.t Lesbegue) and let $W$ be a random $d \times k$ marix with independent entries whose distributions also have density. Let $f:\mathbb R \to \mathbb R$ be a continuous function and conside the $n \times k$ random matrix $Z$ with entries given by $z_{ij} := f(x_i^\top w_j)$. Finally, let $Q := ZZ^\top$ an $n \times n$ random psd matrix.
Question. If $n < k$, under what conditions on $f$ is $Q$ is nonsingular almost-surely ?
Observations
- Conditioned on $W$, the rows of $Z$ are independent.