Fig. 1

Left: Gaussian annulus theorem [30]: for a d-dimensional multi-variant Gaussian with unit variance in all directions, for any \(\beta \le \sqrt {d}\), all but at most \(3e^{-c\beta ^{2}}\) of the probability mass lies within the annulus \(\sqrt {d}-\beta \le \|{x}\| \le \sqrt {d} + \beta \), where c is a fixed positive constant. The color region shown in the figure represents the annulus. Rigid proof can be found in [30]. Right: The length distribution of samples from a 512-dimensional Gaussian distribution N(0,I)