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

TL;DR

This paper derives second-order achievability bounds for indirect quadratic lossy source coding, considering specific source models with bounded functions and moments, extending to cases with joint reconstruction of hidden and observable sources.

Contribution

It introduces new second-order bounds for indirect quadratic lossy source coding under specific source models, generalizing previous finite-alphabet results.

Findings

Established second-order achievability bounds using distortion-tilted information.

Generalized bounds to joint reconstruction of hidden and observable sources.

Bounds are consistent with finite-alphabet counterparts and are second-order tight.

Abstract

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.