When I want to obtain statistical properties of the matrix-variate lasso method, for example, $$ \hat{X}=\underset{X\in \mathbb{R}^{n\times n}}{\arg \min}\mathcal{L}\left(X\right)+\lambda_n \|X\|_1, $$ where $\mathcal{L}$ is the loss function and $\|X\|_1=\sum_{i,j}|X_{ij}|$, I always focus on the KKT condition from its directional derivative. Thus, I have that $$ \mathcal{L}^{\prime}\left(\hat{X}\right)+\lambda_n Z, $$ where $Z\in \partial\|\hat{X}\|_1$ and $\|Z\|_{\max}=\underset{i,j}{\max}|Z_{ij}|\le 1$.
Could we bound the spectral norm of $Z$? I learned that if $Z$ is the subgradient of the nuclear norm, the spectral norm of $Z$ is smaller than 1. Does the element-wise 1 norm have a similar property?
