I have a function and would like to get its convexity or concavity property. Different from normal cross-entropy, here $y_i$ is not one-hot, hence the property is bit different with that perfect case. \begin{equation} -\sum_{j\in\mathbb{C}}y_ilog(p_i)\quad (1) \end{equation}
$p_i$ is softmin. $p_i = \frac{e^{-c_i}}{\sum_{j\in\mathbb{C}}e^{-c_j}}$
According to my own calculation, the Hessian matrix of function (1) is as follows: \begin{matrix} 1-c_1p_1+(c_1-1)p_1^2 & -p_1p_2(c_1+c_2+c_3) & -p_1p_3(c_1+c_2+c_3)\\ -p_2p_1(c_1+c_2+c_3) & 1-c_2p_2+(c_2-1)p_2^2 & -p_2p_3(c_1+c_2+c_3)\\ -p_3p_1(c_1+c_2+c_3) & -p_3p_2(c_1+c_2+c_3) & 1-c_3p_3+(c_3-1)p_3^2 \end{matrix}
if I'd like to find its concavity then there are two ways: 1. prove for all $x$, $x^THx\leq0$, 2. for all eigenvalue, $eigen\leq0$.
I'd like to ask which way should I conduct the property of this function. Which way will be easier on calculation or can use some nice property of softmin.
Could you also attach the calculation process, thank you in advance!
ps: If you find any calculation (e.g. Hessian matrix) I calculated was wrong, please also point it out.