Many problems in computational science require the approximation of high-dimensional functions. Examples of such problems can be found in physics, stochastic analysis or statistical learning. Other examples include parametric or uncertainty analyses for parameter-dependent models.
The approximation of a high-dimensional function requires the introduction of specific approximation formats with a low number of parameters that can be estimated using limited information on the function. In this talk, we will first review some classical tools for high-dimensional approximation. We will then give an introduction to low-rank approximation formats, including tree tensor networks, and make the connection between these formats and a particular family of deep neural networks.
Finally, we will briefly discuss algorithms for the approximation using tree tensor networks and present some approximation results.