  In this largely expository note, we present an impossibility result for inner
product recovery in a random geometric graph or latent space model using the
rate-distortion theory. More precisely, suppose that we observe a graph $A$ on
$n$ vertices with average edge density $p$ generated from Gaussian or spherical
latent locations $z_1, \dots, z_n \in \mathbb{R}^d$ associated with the $n$
vertices. It is of interest to estimate the inner products $\langle z_i, z_j
\rangle$ which represent the geometry of the latent points. We prove that it is
impossible to recover the inner products if $d \gtrsim n h(p)$ where $h(p)$ is
the binary entropy function. This matches the condition required for positive
results on inner product recovery in the literature. The proof follows the
well-established rate-distortion theory with the main technical ingredient
being a lower bound on the rate-distortion function of the Wishart distribution
which is interesting in its own right.
