I would like to prove/disprove that the following function is convex with respect to $w$: $$ \sum_{i=1}^{N}\ln[1+(w^{t}X_iw)^{-y}] $$
where I have N data points, $w$ is the $p\times 1$ vector, $X_{i}$ is a $p \times p$ diagonal matrix and $y$ is either $-1$ or $+1$.
I got stucked with the powered quadratic form, I have no idea how to deal with this. Online materials do not seem to includ the case of the powered quadratic form. Really appreciate any help
It is not convex. Take $N=1$, $p=1$ and $y=-1$. Take $X_1 = 1$. In this case, your function is a one-dimensional function defined by: $$ \ln[1+w^2] $$ It is not convex by the second derivative test. You can also plot it and see visually.