Source Coding and Simulation

源编码（数据压缩、量化）和模拟（从纯随机机制（如掷硬币）中产生具有规定分布的信号）之间的密切联系已经被发现了30多年。最近的研究结果表明，在压缩代码设计、为信号处理和编码系统的分析和设计建模随机过程、以及理解捕获现实世界中产生的信号的重要属性的随机过程的数学模型的性质和结构等相关问题上，它们与潜在的重大意义有着更深的联系。这项研究关注的是发展精确的结果，表征和应用这些联系，以获得新的见解，理论和算法。1977年，一个优化的源编码系统的性能被证明受到源的约束模拟质量的限制，在某些条件下是相等的。2008年，一个信息论的“民间定理”的精确表述和证明加强了这种联系，该定理指出，在接近香农最优的源编码系统产生的比特流在Ornstein的d-bar过程距离的严格接近意义上“几乎是投掷硬币”。总之，这些结果意味着源编码系统——包括数字语音、音频、图像和视频通信和存储系统——和模拟系统——包括固定编码或iid位的滤波——在数学上是近似等效的系统，因此适用于一个系统的理论和设计算法对另一个系统产生相应的结果。本项目利用这些联系，基于已知分布和从数据中学习到的分布，为压缩、仿真和建模的代码开发新的理论和设计算法。

The intimate connection between source coding (data compression,quantization) and simulation (the generation of a signal with prescribed distributions from a purely random mechanism such as coin flips) has been known for over three decades. Recent results suggest even deeper connections with potentially significant implications for the related problems of compression code design, modeling random processes for the analysis and design of signal processing and coding systems, and understanding the nature and structure of mathematical models of random processes capturing the important properties of signals arising in the real world. This research is concerned with developing precise results characterizing and applying these connections to obtain new insight, theory, and algorithms.In 1977 the performance of an optimized source coding system was shown to be bounded by the quality of a constrained simulation of the source, with equality under certain conditions. In 2008 this connection was strengthened by a precise formulation and proof of an information theoretic ``folk theorem'' stating that source coding systems performing near the Shannon optimum yield bit streams that are ``nearly coin flips'' in the rigorous sense of closeness in Ornstein's d-bar process distance. Together these results imply that source coding systems --- including digital speech, audio, image, and video communication and storage systems --- and simulation systems --- comprising stationary codings or filterings of iid bits --- are mathematically approximately equivalent systems, and hence the theory and design algorithms appropriate for one yield corresponding results for the other. This project exploits these connections to develop new theory and design algorithms for codes for compression, simulation, and modeling based on known distributions and on distributions learned from data.