# 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 $$(e_i)_{1\leq i\leq M}$$?

The collection of scalar products is often called measurement. The collection of vectors (each being an atom) can be called measurement dictionaries or sensing dictionaries. This is not specific to random atoms.

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.