A Brief Survey on the Approximation Theory for Sequence Modelling

The modelling of relationships between sequences is an important task that enables a wide array of applications, including classical time-series prediction problems in finance [1], and modern machine learning problems in natural language processing [2]. Another class of engineering applications involving sequential relationships are control systems, which study the dependence of a dynamical trajectory on an input control sequence [3]. In general, sequence-to-sequence relationships can be very complex. For example, when the index set for the sequences is infinite, one can understand these relationships as mappings between infinite-dimensional spaces. Thus, traditional modelling techniques are limited in their efficacy, especially when there is little prior knowledge on the system of interest.