Computing the rate-distortion function for continuous sources is commonly regarded as a standard continuous optimization problem. When numerically addressing this problem, a typical approach involves discretizing the source space and subsequently solving the associated discrete problem. However, existing literature has predominantly concentrated on the convergence analysis of solving discrete problems, usually neglecting the convergence relationship between the original continuous optimization and its associated discrete counterpart. This neglect is not rigorous, since the solution of a discrete problem does not necessarily imply convergence to the solution of the original continuous problem, especially for non-linear problems. To address this gap, our study employs rigorous mathematical analysis, which constructs a series of finite-dimensional spaces approximating the infinite-dimensional space of the probability measure, establishing that solutions from discrete schemes converge to those from the continuous problems.

翻译：计算连续信源的率失真函数通常被视为标准连续优化问题。在数值求解该问题时，典型方法是对信源空间进行离散化并求解相应的离散问题。然而现有文献主要集中于离散问题求解的收敛性分析，往往忽略了原始连续优化问题与其离散近似形式之间的收敛关系。这种忽视显然不够严谨，因为离散问题解未必能保证收敛至原始连续问题解，尤其对于非线性问题而言。为弥补这一研究空白，本研究采用严格数学分析，通过构造逼近概率测度无穷维空间的一系列有限维空间，证明了离散格式解收敛到连续问题解的理论依据。