Tensors in Data Sciences

Tensors have long been used in physics, as they are invariant to the coordinate systems used. They also appear when we want to evaluate the arithmetic complexity of certain problems. Finally, we come across them in statistics, with the moments and cumulants of multivariate random variables. But more recently, tensors have found their way into other sectors of the engineering world, such as telecommunications, biomedical engineering, chemometrics, signal processing and many others. Yet it is not the independence of the coordinate system in their definition that has allowed tensors to return to the heart of the algebraic tools used by engineers. So what happened?

One of the fundamental properties of tensors is that they can be uniquely decomposed into a sum of simpler tensors, under fairly weak assumptions. And in many situations, these simpler terms have an interesting physical meaning. It's this uniqueness that has led to their renewed interest over the past dozen years. We describe this fundamental property in section 2 . However, in the presence of measurement errors (e.g. noise), the measured tensor does not have the expected rank, so the best low-rank approximation should be calculated. However, this approximation does not always exist, as we show in section 3 , which complicates the implementation of algorithms. Finally, several flagship applications are described in the 4