Is sparsity induced penalty in source separation “Entrywise matrix norms”?

Question

I am reading this paper where they introduce norm penalties for source separation. In table 1, the $\log/ l_1$ type is $\sum_{g} log(\epsilon + \lVert H_{g} \rVert_1)$. I wonder this $\lVert H_{g} \rVert_1$ refer to which definition of $l1$ norm? In wikipedia, it has 2 definitions for this symbol.

1. "Entrywise" matrix norms: $\Vert A \Vert_p = \Vert \mathrm{vec}(A) \Vert_p = \left( \sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^p \right)^{1/p}$ (wiki) or

2. Matrix norms induced by vector norms: $ \|A\|_1 = \max_{1 \leq j \leq n} \sum_{i=1}^m | a_{ij} |$ (wiki)

which is simply the maximum absolute column sum of the matrix.

One thing is below the table, the description said ""all norms are elementwise over matrix entries". So I doubt it is the first definition but not fully understand. Very appreciate if you may add explanation why it is that type of norm.

Answer 1

I haven't read the paper, but from the information you provided it appears the appropriate definition is the first one you gave, i.e., the "entrywise" matrix norm.

The text you quoted refers to the norms being "elementwise" which is equivalent to being "entrywise". One way to look at this norm is as the $l_1$ equivalent of the Frobenius norm. Essentially, you are replacing the sum of the squares of the matrix elements with the sum of the absolute value of the elements.

Trying to minimize the $||H_g||_1 $ norm using this definition, is an approximation for trying to reduce the number of non-zero entries in the matrix $H_g$. You would have to examine the problem formulation in the paper to see if that makes sense.