Low-Rank Approximability and Entropy Scaling for Eigenfunctions of a PDE

We want to identify and describe the mathematical structure of "sparsity" in high-dimensional problems. Systems that depend on a large number of variables are known to suffer from the curse of dimensionality: their complexity generally grows exponentially in the number of variables. Nevertheless, decades of research have shown that in many cases such systems can be accurately approximated with polynomial complexity. Perhaps the most studied phenomena in this context are entangled quantum mechanical systems obeying Area Laws. In such cases the information content scales much slower than the size of the system.

In this talk we will discuss the links between entropy Area Laws and low-rank approximation. We will see how a local (NNI) operator admits eigenfunctions with favorable approximation properties. This will lead to an Area Law for PDEs. We will conclude by discussing the necessary assumptions, evaluate the potential and limitations of this approach.