Replacing Gaussian observations in memory-constrained inference

Replacing the Gaussian rows used to select a finite message by independent rows increases the remaining conditional information by at most $Cd$, for a uniform spherical signal, message entropy at most d${}^2$, and the specified row dimensions proportional to d. As an application, we prove that learners with $M=o(d^2)$ persistent bits need $T=\Omega (d\log (1/ϵ))$ exact observations to attain uniform-sphere angular success at least 3/5, for $0<ϵ\le 1/10$ and a deterministic finite horizon.