I have two objective function
$$ \min_f \quad \log (1 + \exp (f(x)^Tf(y) - f(x)^Tf(z))) $$
And
$$ \min_f \quad \max(0, f(x)^Tf(y) - f(x)^Tf(z)) $$
where $f$ is a function with its parameters, $x, y,z$ are some variables. How to show they are equivalent? For exmaple, optimize one will optimize the other.
The question is from this paper, the authors say that their Eq.(4) and Eq.(5) are equivalent. However, They don't provide some details in the paper.
Thanks.
The functions $h(t) = \log(1 + e^t)$ and $g(t) = \max\{0, t\}$ are both strictly increasing when $t \ge 0$. As long as $f(x)^T(f(y) - f(z)) \ge 0$ is guaranteed for all $f$, of course the same function will minimize both. (Note, by the way, that $x, y, z$ are not variables for this calculation - they are fixed.)
The authors only claim this is true under "mild assumptions" on $f$. Presumably these assumptions, along with whatever requirements are placed on $f$ by the definition of the role it plays in the paper, are sufficient to show that $f(x)^T(f(y) - f(z)) \ge 0$.