I have a question regarding convexity of logdet function.
Given the convex set $\mathcal{C}=\{{\bf W}| w_{f,k}\in \mathbb{R}^+,\, \sum_f w_{f,k}^2\leq1\,\,\forall k\}$, is the function $I({\bf W})$ convex/concave in set $\mathcal{C}$. The function is given as $I({\bf W})=log\,det({\bf I}+\sum_f diag\{w_f,.\}{\bf {G^f}^HG^f}diag\{w_f,.\})$ where $\bf G^f$ is a deterministic matrix given for each index $f$, and $diag\{w_f,.\})$ is a diagonal matrix with $f$th column of matrix $\bf W$ i.e, $w_{f,.}$ as diagonal.
The answer is negative. A one-dimensional example is $\mathcal C = \{ w \geq 0 : w^2 \leq 1 \}$, and $G^f = c$, which results in the function $I(w) = \log(1+c^2x^2)$ on $[0,1]$. The second derivative of $I$ is negative for $w>\sqrt{1/c}$, so for $c>1$ the function is not convex on $\mathcal C$.