Rank-Metric in Coding Theory and Machine Learning

两个不同的团体-信息论和机器学习-最近开始研究在仿射空间中找到最小秩矩阵的数学问题。他们这样做是出于完全不同的原因，并提出了非常不同的方法。在机器学习中，秩被认为是一个非常有用的正则化参数，用于解决不适定的逆问题。在过去的四年中，几十个应用thelow-rank恢复问题已被确定。它们的范围从图像处理，从二次测量信号的鲁棒恢复，到从不完整数据预测在线商店的用户偏好。 独立地，研究编码理论的研究人员已经意识到，在某些网络编码场景中自然发生的错误在表示为合适的矩阵时是低秩的。解码问题在形式上等价于机器学习中的低秩恢复问题。(This类似于压缩感测和汉明度量解码之间的关系，这在过去已经被富有成效地利用）。尽管这两项任务非常相似，但迄今为止，两个社区之间几乎没有任何概念和方法的转移。这个项目联合了两个分别在编码和低秩恢复方面有专长的小组，旨在改变这种情况。

Two different communities - information theory and machine learning - have recently started to investigate the mathematical problem of finding the matrix of minimal rank in an affine space. They have done so for completely different reasons and have proposed very different approaches. In machine learning, rank has been identified as an extremely useful regularization parameter for otherwise ill-posed inverse problems. In the past four years, dozens of applications of thelow-rank recovery problem have been identified. They range from image processing, over robust recovery of signals from quadratic measurements, to the prediction of user preferences in online shops from incomplete data. Independently, researchers working on coding theory have realized that errors that naturally occur in certain network coding scenarios are of low rank when represented as suitable matrices. The decoding problem is formally equivalent to thelow-rank recovery one of machine learning. (This is analogous to the relation between compressed sensing and Hamming-metric decoding, that has been fruitfully exploited in the past). Despite the close resemblance between the two tasks, almost no transfer of concepts and methods between the two communities has taken place so far. This project - uniting two groups with expertise in, respectively, coding and low-rank recovery - aims to amend this situation.