I came across the following notation in a paper and am wondering what it means
$$\mathbb{E}\Vert\textbf{u}_i\Vert^2_{\textbf{P}_i}.$$
where $\textbf{u}_i$ is a vector and ${\textbf{P}_i}$ is a matrix. What is the matrix subscript referring to? I only have one page of this paper and have no idea where it came from, so I don't know if this has been specifically defined for this work or if it is accepted notation that I am not aware of.
If the $P_i$ are positive definite, $\|x\|_{P_i}^2$ may be either $$ x^T P_i x\quad\text{or}\quad x^TP_i^{-1}x. $$