I have two vectors ${\bf x} \in \mathbb{R}^d$ and ${\bf y} \in \mathbb{R}^p$ such that $d < p$ and let ${\bf A} \in \mathbb{R}^{s\times d}$ and ${\bf B} \in \mathbb{R}^{s\times p}$two matrices, where $s < d$. I'm looking for an interpretation or terminology (if it exists) to describe the role of ${\bf A}$ and ${\bf B}$ when comparing ${\bf x}$ and ${\bf y}$ using $\|{\bf A} {\bf x}- {\bf B} {\bf y}\|$ where $\|\cdot\|$ is the Euclidean norm. Since ${\bf A}$ and ${\bf B}$ are not a square matrices, then I cannot considered them to be projections. From reading other questions, this could be related to the matrix of a linear map but not sure if this is the only interpretation we can have.
I have the following naïve example to illustrate my point: ${\bf x}$ and ${\bf y}$ could be measurements from two sensors having in common $s$ features. So, I want to compare both vectors on the common space $\mathbb{R}^s$.