Lower-bound for smallest eigenvalue of random $k \times $k matrix $C(W)$ defined by $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$

Question

Let $k$ and $d$ be positive integers such that $d/k:=\lambda > 1$. Let $W$ be $k \times d$ random matrix with rows $w_1,\ldots,w_k \in \mathbb R^d$ drawn iid from $N(0,(1/d)I_d)$, and define the $k \times k$ matrix $C(W)$ by setting $C(W)_{i,j} := 2(w_i^\top w_j)^2 + \|w_i\|^2\|w_j\|^2$.

Question. Is there a high-probability good lower-bound for the smallest eigenvalue of $C(W)$ ?

N.B. I'm familiar with standard RMT.

Answer 1

We have $$ C(W) = 2 A \circ A + v v^\top$$ where $v$ is the vector with entries $\|w_i\|^2$, $A$ is the Wishart matrix with entries $w_i^\top w_j$, and $\circ$ is the Hadamard product. From the Schur product theorem (and the fact that adding a positive semi-definite matrix to a self-adjoint matrix only serves to increase the least eigenvalue $\lambda_1$) we conclude that $$ \lambda_1(C(W)) \geq 2 \lambda_1(A)^2.$$ Combining this with the Marchenko-Pastur law one gets an almost sure lower bound of $2 (1 - 1/\sqrt{\lambda})^4 - o(1)$ (if I have not made a sign error). However, the application of the Schur product theorem may be inefficient and one could hope to improve this lower bound slightly.

Note from the Weyl inequalities that $$ 2\lambda_1(A \circ A) \leq \lambda_1(C(W)) \leq 2 \lambda_2(A \circ A) $$ so the problem really boils down to understanding the spectral behavior of the Hadamard square $A \circ A$ of a Wishart matrix $A$.