# What do you call the random Gaussian vectors in compressed sensing?

Let $$(e_i)_{1\leq i\leq M}$$ be vectors with zero-mean i.i.d. entries from a Gaussian distribution. In a compressed-sensing setup, I have observed the collection of scalar products $$(\langle x\, |\, e_i\rangle)_{1\leq i\leq M}$$ for an unknown vector $$x$$.

In the compressed sensing literature, what is the name for the collection of vectors $$(e_i)_{1\leq i\leq M}$$?

In most of the literature I am familiar with (signal processing), these vectors are considered collectively as rows in a matrix, i.e. $$\mathbf y = \mathbf E \mathbf x$$ where the $$(M \times 1)$$ vector $$\mathbf y$$ contains the measurements, i.e. the inner products of the Gaussian vectors $$\mathbf e_i$$ with $$\mathbf x$$. The sparse vector $$\mathbf x$$ is $$(N \times 1)$$. NB, I prefer to use bold notation for vectors and matrices.
Hence this $$(M \times N)$$ matrix collection of vectors is called the maesurement matrix.
I am reluctant to call it a measurement dictionary as in this answer, because in a lot of literature, the matrix that takes you from the sparse vector $$\mathbf x$$ to the measurements $$\mathbf y$$ is factored into a measurement matrix (here $$\mathbf E$$) and a dictionary matrix or sparsifying dictionary $$\mathbf D$$ - typically $$(N \times N)$$ matrix: $$\mathbf y = \mathbf{EDx} = \mathbf{Ax}$$ This factorisation is especially important in applications where the compressed sensing equation represents a physical phenomenon where the observable signal $$\mathbf z = \mathbf{Dx}$$ is not necessarily sparse in the observable domain, but the measurement $$\mathbf y = \mathbf {Ez}$$ is the only way you physically have acces to measuring your signal. The interpretation $$\mathbf z = \mathbf{Dx}$$ is a mathematical abstraction you add to your model afterwards in order to apply compressed sensing to the reconstruction of the signal.