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

For an i.i.d. Gaussian source with variance σ ², we show that it is necessary to spend 1/2 ln σ ²/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).