Notation

As the context and convention permits, small Roman and Greek letters are used for scalars, small Roman boldface letters for vectors, capital Roman boldface for matrices, and capital Greek for functions. Letters early in the alphabet tend to indicate constants and letters towards the end indicate variables. Capital caligraphs indicate sets and $\vert\bullet\vert$ their cardinality. $\vert\bullet\vert$ is the $L_2$ norm and bracketed super-scripts are indices rather than the power function. The letters $K$, $N$, $D$ are in general scalars and refer to particular versions of $k$, $n$, $d$, respectively. $\mathbf{I}$ denotes the identity matrix, $\mathbf{0}$ and $\mathbf{1}$ are the vector / matrix equivalents to 0 and 1. $O(\bullet)$ is the Landau symbol. For a $d \times n$ matrix $\mathbf{X}$, $x_{i,j}$ refers to the element at the $i$-th row and $j$-th column and $\mathbf{x}_j$ ( $\mathbf{x}_{\dagger i}$) is the $j$-th $d \times 1$ column ( $n \times 1$ row) vector of $\mathbf{X}$. $\mathbf{X}^{\dagger}$ denotes the transpose of $\mathbf{X}$. The set of real numbers is $\mathbb{R}$, the set of non-negative (strictly positive) real numbers is $\mathbb{R}_{+}$ ( $\mathbb{R}_{++}$).