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Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random matrix, independent of $X$, with entries drawn iid from $N(0,1/d)$. Consider the $n \times k$ matrix $C$ defined by $c_{ij} = \max(x_i^\top w_j, 0)$.

Question. What are good bounds for the extreme singular-values of $C$ ?

Related: https://math.stackexchange.com/q/4031609/168758

Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random matrix, independent of $X$, with entries drawn iid from $N(0,1/d)$. Consider the $n \times k$ matrix $C$ defined by $c_{ij} = \max(x_i^\top w_j, 0)$.

Question. What are good bounds for the extreme singular-values of $C$ ?

Empirical observations

Related: https://math.stackexchange.com/q/4031609/168758

Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random matrix, independent of $X$, with entries drawn iid from $N(0,1/d)$. Consider the $n \times k$ matrix $C$ defined by $c_{i,j} = \max(x_i^\top w_j, 0)$.

Question. What are good bounds for the extreme singular-values of $C$ ?

Related: https://math.stackexchange.com/q/4031609/168758

Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random matrix, independent of $X$, with entries drawn iid from $N(0,1/d)$. Consider the $n \times k$ matrix $C$ defined by $c_{ij} = \max(x_i^\top w_j, 0)$.

Question. What are good bounds for the extreme singular-values of $C$ ?

Related: https://math.stackexchange.com/q/4031609/168758

Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random matrix, independent of $X$, with entries drawn iid from $N(0,1/d)$. Let $\psi$ $1$-Lipschitz and consider the $n \times k$ matrix $C$ defined by $c_{i,j} = \max(x_i^\top w_j, 0)$.

Question. What are good bounds for the extreme singular-values of $C$ ?

Related: https://math.stackexchange.com/q/4031609/168758

Let $n,d,k$ be large positive integers such that $\max(n/d,k/d) =: \lambda < 1$. Let $X$ be a random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $W$ be a $k \times d$ random matrix, independent of $X$, with entries drawn iid from $N(0,1/d)$. Consider the $n \times k$ matrix $C$ defined by $c_{i,j} = \max(x_i^\top w_j, 0)$.

Question. What are good bounds for the extreme singular-values of $C$ ?

Related: https://math.stackexchange.com/q/4031609/168758