Uniform Distribution on $(n-1)$-Sphere: Rate-Distortion under Squared Error Distortion

This paper investigates the rate-distortion function, under a squared error distortion $D$, for an $n$-dimensional random vector uniformly distributed on an $(n-1)$-sphere of radius $R$. First, an expression for the rate-distortion function is derived for any values of $n$, $D$, and $R$. Second, two types of asymptotics with respect to the rate-distortion function of a Gaussian source are characterized. More specifically, these asymptotics concern the low-distortion regime (that is, $D\to 0$) and the high-dimensional regime (that is, $n\to \mathrm{\infty }$).