Rank-Metric in Coding Theory and Machine Learning

点击翻译按钮获取中文摘要

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.