This paper studies the joint data and semantics lossy compression problem, i.e., an extension of the hidden lossy source coding problem that entails recovering both the hidden and observable sources. We aim to study the nonasymptotic and second-order properties of this problem, especially the converse aspect. Specifically, we begin by deriving general nonasymptotic converse bounds valid for general sources and distortion measures, utilizing properties of distortion-tilted information. Subsequently, a second-order converse bound is derived under the standard block coding setting through asymptotic analysis of the nonasymptotic bounds. This bound is tight since it coincides with a known second-order achievability bound. We then examine the case of erased fair coin flips (EFCF), providing its specific nonasymptotic achievability and converse bounds. Numerical results under the EFCF case demonstrate that our second-order asymptotic approximation effectively approximates the optimum rate at given blocklengths.

翻译：本文研究联合数据与语义有损压缩问题，即隐藏有损信源编码问题的扩展，其目标是在恢复隐藏信源的同时恢复可观测信源。我们旨在探究该问题的非渐近与二阶性质，特别是逆界方面。具体而言，我们首先利用失真倾斜信息的性质，推导了适用于一般信源与失真度量的通用非渐近逆界。随后，通过对非渐近逆界进行渐近分析，在标准分组编码框架下推导出二阶逆界。该逆界具有紧致性，因其与已知的二阶可达界一致。接着，我们考察擦除公平硬币翻转案例，并给出其特定的非渐近可达界与逆界。在该案例下的数值结果表明，我们的二阶渐近近似能有效逼近给定分组长度下的最优压缩速率。