One-shot channel simulation is a fundamental data compression problem concerned with encoding a single sample from a target distribution $Q$ using a coding distribution $P$ using as few bits as possible on average. Algorithms that solve this problem find applications in neural data compression and differential privacy and can serve as a more efficient alternative to quantization-based methods. Sadly, existing solutions are too slow or have limited applicability, preventing widespread adoption. In this paper, we conclusively solve one-shot channel simulation for one-dimensional problems where the target-proposal density ratio is unimodal by describing an algorithm with optimal runtime. We achieve this by constructing a rejection sampling procedure equivalent to greedily searching over the points of a Poisson process. Hence, we call our algorithm greedy Poisson rejection sampling (GPRS) and analyze the correctness and time complexity of several of its variants. Finally, we empirically verify our theorems, demonstrating that GPRS significantly outperforms the current state-of-the-art method, A* coding. Our code is available at https://github.com/gergely-flamich/greedy-poisson-rejection-sampling.

翻译：一次性信道模拟是一个基础性的数据压缩问题，其目标在于使用编码分布 $P$ 对目标分布 $Q$ 中的单个样本进行编码，并尽可能减少平均所需的比特数。解决该问题的算法可应用于神经数据压缩和差分隐私领域，并可作为基于量化方法的一种更高效的替代方案。遗憾的是，现有解决方案速度过慢或适用性有限，阻碍了其广泛采用。在本文中，我们针对目标-提议密度比为单峰的一维问题，描述了一种具有最优运行时间算法，从而彻底解决了该场景下的一次性信道模拟。我们通过构造一个等价于在泊松过程点上进行贪心搜索的拒绝采样过程来实现这一点。因此，我们将该算法命名为贪心泊松拒绝采样（GPRS），并分析了其多种变体的正确性和时间复杂度。最后，我们通过实验验证了理论结果，表明GPRS显著优于当前最先进的方法A*编码。我们的代码见https://github.com/gergely-flamich/greedy-poisson-rejection-sampling。