On the Second-Order Achievabilities of Indirect Quadratic Lossy Source Coding

This paper studies the second-order achievabilities of indirect quadratic lossy source coding for a specific class of source models, where the term "quadratic" denotes that the reconstruction fidelity of the hidden source is quantified by a squared error distortion measure. Specifically, it is assumed that the hidden source $S$ can be expressed as $S = \varphi(X) + W$, where $X$ is the observable source with alphabet $\mathcal{X}$, $\varphi(\cdot)$ is a deterministic function, and $W$ is a random variable independent of $X$, satisfying $\mathbb{E}[W] = 0$, $\mathbb{E}[W^2] > 0$, $\mathbb{E}[W^3] = 0$, and $\mathbb{E}[W^6] < \infty$. Additionally, both the set $\{\varphi(x):\ x \in \mathcal{X} \}$ and the reconstruction alphabet for $S$ are assumed to be bounded. Under the above settings, a second-order achievability bound is established using techniques based on distortion-tilted information. This result is then generalized to the case of indirect quadratic lossy source coding with observed source reconstruction, where reconstruction is required for both the hidden source $S$ and the observable source $X$, and the distortion measure for $X$ is not necessarily quadratic. These obtained bounds are consistent in form with their finite-alphabet counterparts, which have been proven to be second-order tight.