Consider $J$ a random matrix of size $n\times n$ with i.i.d. Gaussian entries $J_{ij} \sim \mathcal{N}(0,\sigma^2/n)$. Let $f(x)=tanh(x)$, and for $x\in\mathbb{R}^n$, $f(x)$ denotes the vector where $f(.)$ is applied entry-wise.
My goal is to compute
$$M_n(\sigma) = \inf_{K\in \mathbb{R}^{n\times n}} \sup_{x\in\mathbb{R}^n} \frac{||KJf(x)||}{||Kx||}$$
In particular, how $M_n(\sigma)$ compares with $\sigma$ for $n$ large ?
