What means a formula with ∞1?

Question

In the paper https://arxiv.org/abs/1602.04938 there is a formula

What does the formula for $\Omega(g)$ mean? I understand that $w_g$ is a vector and that, using the zero norm $$||w_g||_0 > K$$ means "the number of non-zero elements of $w_g$ must be greater than $K$", but I don't get the rest.

Answer 1

The $\mathbb{1}$ means the indicator function, which is used here to say that it should take the value 0 when the condition inside is false, and the value 1 when the condition inside is true. That is, as a function of $g$, we have $$ \mathbb 1[\|w_g\|_0 > K] = \begin{cases} 1 & \text{if $\|w_g\|_0 > K$,}\\ 0 & \text{otherwise.} \end{cases} $$ The infinity symbol here just means "infinity" (some value larger than all real numbers), and the fact that they are written next to each other just means "multiplication", where it is supposed to be understood that $\infty \times 1 = \infty$, and $\infty \times 0 = 0$. Thus the entire function is $$ \Omega(g) = \infty \times \mathbb 1[\|w_g\|_0 > K] = \begin{cases} \infty & \text{if $\|w_g\|_0 > K$,}\\ 0 & \text{otherwise.} \end{cases} $$ (None of this is standard by the way, and it makes sense that you're confused.)

Answer 2

My guess $$ \mathbb 1 [\cdots] $$ means the indicator function of $[\cdots]$, so that formula for $\Omega(g)$ means: $\infty$ on the set $[\cdots]$ and $0$ otherwise.

Answer 3

I believe it means $\Omega(g)$ is infinity if $w_g $ exceeds $K$ and zero otherwise. That would be consistent with implementing a hard constraint as a cost function.