I have the following optimization
$$\min_X f(X)+\lambda \|X\|_1,$$ in which $X\in \mathbb{R}^{n \times k}$, and $\|X\|_1=\sum_{i,j} |x_{ij}|$, and $f(X)$ is differentiable in $X$.
It is ideal for me to have a small value for $\|X\|_1$ which is equally balanced over the columns of $X$.
For example, I like to have $\|X_{:,i}\|_1 \approx \|X_{:,j}\|_1~\forall i,j$, where $X_{:,i}$ denotes colimn $i$ of $X$.
Specifically, I want to prevent situations in which $X$ has a few columns with large entries and a lot of columns with very small entries, assuming $f(X)$ allows those situations.