Fixed-length lossy compression in the finite blocklength regime: Gaussian source

For an i.i.d. Gaussian source with variance σ², we show that it is necessary to spend ½ ln σ²/d + 1/√(2n) Q^-1(ε) + O (ln n/n) nats per sample in order to reproduce n source samples within mean-square error d with probability at least 1 - ε, where Q^-1 (·) is the inverse of the standard Gaussian complementary cdf. The first-order term is the rate-distortion function of the Gaussian source, while the second-order term measures its stochastic variability. We derive new achievability and converse bounds that are valid at any blocklength and show that the second-order approximation is tightly wedged between them, thus providing a concise and accurate approximation of the minimum achievable source coding rate at a given fixed blocklength (unless the blocklength is very small).