In the theory of lossy compression, the rate-distortion (R-D) function $R(D)$ describes how much a data source can be compressed (in bit-rate) at any given level of fidelity (distortion). Obtaining $R(D)$ for a given data source establishes the fundamental performance limit for all compression algorithms. We propose a new method to estimate $R(D)$ from the perspective of optimal transport. Unlike the classic Blahut--Arimoto algorithm which fixes the support of the reproduction distribution in advance, our Wasserstein gradient descent algorithm learns the support of the optimal reproduction distribution by moving particles. We prove its local convergence and analyze the sample complexity of our R-D estimator based on a connection to entropic optimal transport. Experimentally, we obtain comparable or tighter bounds than state-of-the-art neural network methods on low-rate sources while requiring considerably less tuning and computation effort. We also highlight a connection to maximum-likelihood deconvolution and introduce a new class of sources that can be used as test cases with known solutions to the R-D problem.

翻译：在无损压缩理论中，率失真函数$R(D)$描述了在任意给定保真度（失真）水平下，数据源可被压缩的程度（以比特率表示）。针对特定数据源获取$R(D)$，确立了所有压缩算法的基本性能极限。本文从最优传输视角提出一种估计$R(D)$的新方法。与经典Blahut-Arimoto算法预先固定再生分布支撑集不同，我们的Wasserstein梯度下降算法通过移动粒子学习最优再生分布的支撑集。我们证明了该算法的局部收敛性，并基于与熵最优传输的联系，分析了率失真估计器的样本复杂度。实验表明，在低速率数据源上，我们获得了与最先进神经网络方法相当或更紧的界限，同时显著减少了调参和计算开销。此外，我们揭示了该方法与最大似然反卷积的关联，并引入了一类可解析求解率失真问题的新数据源作为测试案例。