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,
$D\rightarrow 0$
) and the high-dimensional regime (that is,
$n\rightarrow\infty$
).