# What does it mean for all norms are elementwise over matrix entries

I am reading this paper where they introduce 4 types of norm penalties which induce sparsity in the matrix block $H_g$ in table 1. For example, $\log/ l_1$ is $\sum_{g} log(\epsilon + \lVert H_{g} \rVert_1)$.

As the definition of $\lVert H_{g} \rVert_1$, it is the maximum absolute column sum of the matrix (wiki). But in the paper, they give a description under the table "all norms are elementwise over matrix entries". How can norm are elementwise? I think from the definition, it seems to be columnwise.

• it just means that you consider the $\ell_1$ norm of the "vectorized" form of matrix, i.e., in this case, $\|H_g\|_1 = \sum_{i,j} |(H_g)_{i,j}|$. Commented Apr 9, 2018 at 7:24
• @Onetimething So it is "Entrywise matrix norms" wiki1 Not "Matrix norms induced by vector norms" wiki2?
– Jan
Commented Apr 9, 2018 at 7:29
• Yes, that is correct. The maximum column sum is the induced vector norm. Commented Apr 9, 2018 at 7:30
• @Onetimething May you explain a bit while it is that type of norm?
– Jan
Commented Apr 9, 2018 at 7:46
• I am not entirely sure what you mean, but here are some ideas. (a) in general elementwise $1$-norm is used to promote sparsity (if suppose, you minimize this quantity) in matrices just as is done for vectors. (b) using the definition of induced vector norms, the wiki page you linked to has the derivation to show that the induced $1$-norm is indeed the absolute maximum column sum. Hope this helps. Commented Apr 9, 2018 at 7:56