Collaboration with Yaming Yu - entropy inequalities and thinning

熵量化了一种方式，例如，投掷一枚公平硬币的结果比投掷一枚有偏见的硬币更难预测。它在理解信息如何在嘈杂的通信网络中传输，以及如何在尽可能小的设备中存储大量信息（数据压缩）方面发挥着基础作用。熵是信息论领域的研究对象，该项目旨在利用纯数学各个领域的技术，解决两个主要的突出猜想。这些猜想描述了随机事件和的熵。这是一个熟悉的事实，虽然随机性无法预测，但通过对随机事件求和或平均，不可预测性就会被抵消。因此，例如，虽然一次抛硬币是不可能预测的，但我们可以确信，在1,000,000次公平抛硬币中，将有498,000到502,000个正面。这个项目将考虑熵在这种情况下的行为，帮助理解这种影响。更具体地说，许多信息传输模型假设，由于发送者或接收者无法控制的物理过程，噪声被添加到信号中。最终目的是对接收到的消息进行清理，即去除噪声，以便接收到发送器发送的完整内容。如果我们能够证明上面提到的猜想，就会对某些通信网络产生影响，包括在互联网上实现更高质量的直播视频。例如，Stankovic描述了一种模型，在这种模型中，视频通过无线链路传播到许多服务器，这些服务器在本地压缩信息。最有效的算法是基于所谓的熵权不等式，我们在这里试图扩展它。

Entropy quantifies the way in which, for example, the outcome of tossing a fair coin is harder to predict than with a biased one. It plays a fundamental role in understanding how information is transmitted over noisy communication networks, and how large amounts of information can be stored in as small devices as possible (data compression). Entropy is studied in the field of information theory, and this project aims to resolve two major outstanding conjectures, using techniques from various areas of pure mathematics. These conjectures describe the entropy of the sums of random events. It is a familiar fact that while randomness cannot be predicted, by summing or averaging random events, the unpredictability cancels out. Thus, for example, while a single coin toss is impossible to predict, we can be confident that in 1,000,000 fair coin tosses, there will be between 498,000 and 502,000 heads. This project will consider the behaviour of the entropy in such settings, helping understand such effects.More specifically, many models of information transmission suppose that noise is added to the signal, due to physical processes beyond the control of transmitter or receiver. The ultimate aim is to clean up the received message, that is to remove the noise, in order to receive the full content sent by the transmitter. If we are able to prove the conjectures mentioned above, there would be implications in certain communications networks, including enabling a better quality of live streaming video on the Internet. For example, Stankovic describes a model where such video is broadcast via a wireless link to a number of servers, which compress the information locally. The algorithms which work best are based on the so-called Entropy Power Inequality, which we seek to extend here.