Equation conversion

Question

While reading a paper, I encountered this equation:

Where

The term $f^TLf$ can be written as:

Then first equation is rewritten as :

The point I do not understand is how arg max is converted to arg min ? Shouldn't it be still arg max?

Thank you for an explanation.

Answer 1

Notice that $\max_{S} f = -\min_{S} f$. This holds for $\arg \max$ and $\arg \min$.

Answer 2

Yes, it should be argmax (since no sign change happens during rewriting the objective).

Since $\|f\|_{0}$ represents the number of non zero entries in $f$, and $f\in\{0,1\}^{n}$, \begin{equation} \eta\|f\|_{0}=\sum_{p=1}^{n}\eta f_{p}, \end{equation}

which means that the objective can be rewritten as

\begin{align} c^{\top}f-\lambda f^{\top}Lf-\eta\|f\|_{0}=\sum_{p=1}^{n}(c_{p}-\eta)f_{p}-\lambda\sum_{p\sim q}(f_{p}-f_{q})^{2}. \end{align}