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I have a question concerning the arg min operator. I know how it works for a single 'input' (i.e. input for which the function attains its minimal value), but can it have two inputs? If I have a matrix D, $\mathcal{D}$, X and A, and I have the following equation

$\text{arg min}_{D \in \mathcal{D}, A} ||X-DA||^2_F + \lambda\Omega(A)$,

Does this mean that arg min has two inputs, one for $D \in \mathcal{D}$ and one for the matrix A, or is $D \in \mathcal{D}$ and $D \in A$?

And as a side question, what does the F in $||X-DA||^2_F$ mean when it is subscripted like that? The paper I have it from does not mention anything about F, so I figure there is some basic math knowledge I am missing

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The way I read it is to find the minimum of your expression when $D\in\mathcal D$ and $A$ is unrestricted (your first interpretation). $F$ should be Frobenius norm, but without looking at the paper it's impossible to be sure.

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    Thanks! The paper is 'Learning Word Representations with Hierarchical Sparse Coding' and the equation can be seen on page 2 (Equation 1) – 2017-01-16
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    I looked at the paper and I can confirm that $\lVert\cdot\rVert_F$ is the Frobenius norm. – 2017-01-16
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    Thank you so much. I have somehow managed to confuse myself. The 2 in $||X-DA||^2_F$, it is something like matrix L2 norm? Or am I missing something again? – 2017-01-17
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    The $2$ means that the norm is squared. – 2017-01-17