fa.functional analysis – Lower-bound on $lambda_{min}(C(W)) / lambda_{max}(WW^T)$, for $W in mathbb R^{k times d}$ and $C(W) := Cov_x(psi(Wx))$

Let $x$ be uniformly-distributed on the unit-sphere in $mathbb R^d$, and let $psi:mathbb R to mathbb R$ be a Lipschitz-continuous differentiable function. Given $W=(w_1,ldots,w_k) in mathbb R^{k times d}$, let $C = C(W)in mathbb R^{k times k}$ be the covariance matrix of the random vector $(psi(x^top w_1),ldots,psi(x^top w_k))$, ie.

$$
c_{i,j} := mathbb E_x((psi(x^top w_i)-mu_i)(psi(x^top w_j)-mu_j)),
$$

where $mu_i := mathbb E_x(psi(x^top w_i))$ for all $i in (k)$.

Question. How would one proceed to lower-bound the quantity $inf_{W in mathbb R^{k times d}} alpha(W) := dfrac{lambda_{min}(C(W))}{lambda_{max}(WW^top)}$ ?

My attempt so far

By leveraging Theorem 2.1 of this paper, I’m able to prove the following weaker result:

  • $k asymp d to infty$
  • $psi(t):=max(t,0)$,
  • $W$ is a random matrix with independent entries from $N(0,1)$,

then
$d cdot alpha(W) to Omega(1)$ in probability.