Designing universal causal deep learning models: The geometric (Hyper)transformer

Several problems in stochastic analysis are defined through their geometry, and preserving that geometric structure is essential to generating meaningful predictions. Nevertheless, how design principled deep learning (DL) models capable of encoding these structures remains largely unknown. We address this open problem by introducing a universal causal DL framework which the user specifies suitable pair metric spaces X $\mathcal {X}$ Y {Y}$ our returns model causally approximating any “regular” map sending time series Z {X}^{\mathbb {Z}}$ {Y}^{\mathbb while respecting forward flow information throughout time. Suitable geometries on include various (adapted) Wasserstein arising optimal stopping problems, variety statistical manifolds describing conditional distribution continuous-time finite state Markov chains, all Fréchet admitting Schauder basis, for example, as classical finance. compact subsets Euclidean space. Our results quantitatively express number parameters needed achieve given approximation error function target map's regularity both . Even when omitting temporal structure, theorems first guarantees Hölder functions, between such can be approximated models.