Subsphere methods for memory-sample lower bounds in noiseless Gaussian regression

Let a finite-state streaming learner estimate a uniformly random unit vector from independent exact Gaussian linear measurements. We prove that $o(d^2)$ bits of persistent memory and angular success probability at least 2/3 require at least $2^{-16}d{\log }_2(1/ϵ)$ samples for all sufficiently large d, uniformly for $0<ϵ\le 1/10$. The proof conditions each batch on its observed projection and controls the resulting random residual subsphere.