The function is $\frac{1}{1+\|x_i-c_{yi}\|_2^2}$. Following my computation of the gradient w.r.t. $x_i$. Can somebody please check if it is correct. Let $u = 1+||x_i-c_{y_j}||_2^2$, thus $f = u^{-1}$.
\begin{gather} = \frac{\partial u^{-1}}{\partial u}\frac{\partial }{\partial x_i}({1+||x_i-c_{y_j}||_2^2}) \\ \frac{\partial (u^{-1})}{\partial u} = \frac{-1}{u^2} \\ \frac{\partial }{\partial x_i}(1+||x_i-c_{y_j}||_2^2) = 2{(x_i-c_{y_j})} \end{gather} Substituting these values, the final equation becomes $\frac{-1}{u^2}(2(x_i-c_{y_j}))$. We can simplify this equation to obtain the final gradient which is: $$= \frac{-2(x_i-c_{y_j})}{(1+(x_i-c_{y_j})^2)^{2}}$$ Can anybody please confirm if the calculation is mathematically sound? Also, since the Hessian would be $\Delta^2f(x)<0$, so this is a concave function. Am I correct?
Thanks.