Uniform Distribution on $(n-1)$-Sphere: Rate-Distortion under Squared Error Distortion
Alex Dytso, Martina Cardone
TL;DR
We study the rate-distortion function for a random vector uniformly distributed on the $(n-1)$-sphere of radius $R$ under squared-error distortion. The authors derive an exact non-asymptotic RDF: $\mathsf{R}_n(D;R)=\log(S_{n-1})-h_{\frac{n}{2}}(\sqrt{1-D/R^2})$, with an equivalent integral form $\mathsf{R}_n(D;R)=\int_{0}^{\sqrt{1-D/R^2}} f_{\frac{n}{2}}^{-1}(u)\,du$, and establish a Gaussian proximity bound $\frac{n-1}{n}\mathsf{R}_n^{G}(D;R^2/n)\le\mathsf{R}_n(D;R)\le\mathsf{R}_n^{G}(D;R^2/n)$. In the low-distortion regime, the information dimension is $\mathsf{d}(\mathsf{X}_R)=1-\frac{1}{n}$ for $n>1$ (with $\mathsf{d}=0$ when $n=1$), and as $D\to0^+$ the RDF approaches $\log(2)$ only for $n=1$ while diverging for $n>1$. In the high-dimensional regime, with radius scaling $\sqrt{\alpha_n n}$, the RDF ratio to the Gaussian RDF satisfies $\lim_{n\to\infty}\frac{\mathsf{R}_n(D;\sqrt{\alpha_n n})}{\mathsf{R}_n^{G}(D)}=1+\lim_{n\to\infty}\frac{\log\alpha_n}{\log n}$, implying Gaussian-like behavior when $\log\alpha_n/\log n\to0$. These results illuminate spherical quantization and directional-statistics connections, showing precise RDF expressions and asymptotics for sphere-supported sources that approximate Gaussian behavior in high dimensions.
Abstract
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, $n \to \infty$).