The rate-distortion function (RDF) has long been an information-theoretic benchmark for data compression. As its natural extension, the indirect rate-distortion function (iRDF) corresponds to the scenario where the encoder can only access an observation correlated with the source, rather than the source itself. Such scenario is also relevant for modern applications like remote sensing and goal-oriented communication. The iRDF can be reduced into a standard RDF with the distortion measure replaced by its conditional expectation conditioned upon the observation. This reduction, however, leads to a non-trivial challenge when one needs to estimate the iRDF given datasets only, because without statistical knowledge of the joint probability distribution between the source and its observation, the conditional expectation cannot be evaluated. To tackle this challenge, starting from the well known fact that conditional expectation is the minimum mean-squared error estimator and exploiting a Markovian relationship, we identify a functional equivalence between the reduced distortion measure in the iRDF and the solution of a quadratic loss minimization problem, which can be efficiently approximated by neural network approach. We proceed to reformulate the iRDF as a variational problem corresponding to the Lagrangian representation of the iRDF curve, and propose a neural network based approximate solution, integrating the aforementioned distortion measure estimator. Asymptotic analysis guarantees consistency of the solution, and numerical experimental results demonstrate the accuracy and effectiveness of the algorithm.

翻译：率失真函数长期以来一直是数据压缩的信息论基准。作为其自然扩展，间接率失真函数对应编码器仅能访问与信源相关的观测值而非信源本身的场景。此类场景在遥感和面向目标的通信等现代应用中同样具有重要意义。间接率失真函数可转化为标准率失真函数，其中失真度量被其以观测值为条件的条件期望所替代。然而，当仅给定数据集而需要估计间接率失真函数时，这种转化会带来非平凡挑战：由于缺乏信源与其观测值联合概率分布的统计知识，条件期望无法被计算。为应对此挑战，我们从条件期望即最小均方误差估计器这一已知事实出发，利用马尔可夫关系，确立了间接率失真函数中转化失真度量与二次损失最小化问题解之间的函数等价性，该问题可通过神经网络方法高效逼近。我们进而将间接率失真函数重构为对应其曲线拉格朗日表示的变分问题，并提出基于神经网络的近似解法，整合了前述失真度量估计器。渐近分析保证了解的一致性，数值实验结果验证了算法的准确性与有效性。