Neural networks in non-metric spaces

By leveraging the infinite-dimensional neural network architecture, we proposed in [F. Benth, N. Detering and L. Galimberti, Neural networks in Fréchet spaces, Ann. Math. Artif. Intell. 91 (2023) 75–103] and which can process inputs from Fréchet spaces, and using the universal approximation property shown therein, we now largely extend the scope of this architecture by proving several universal approximation theorems for a vast class of input and output spaces. More precisely, the input space [Formula: see text] is allowed to be a general topological space satisfying only a mild condition (“quasi-Polish”), and the output space can be either another quasi-Polish space [Formula: see text] or a topological vector space [Formula: see text]. Similarly to [F. Benth, N. Detering and L. Galimberti, Neural networks in Fréchet spaces, Ann. Math. Artif. Intell. 91 (2023) 75–103], we show furthermore that our neural network architectures can be projected down to “finite-dimensional” subspaces with any desirable accuracy, thus obtaining approximating networks that are easy to implement and allow for fast computation and fitting. The resulting neural network architecture is therefore applicable for prediction tasks based on functional data. To the best of our knowledge, this is the first result which deals with such a wide class of input/output spaces and simultaneously guarantees the numerical feasibility of the ensuing architectures. Finally, we prove an obstruction result which indicates that the category of quasi-Polish spaces is in a certain sense the correct category to work with if one aims at constructing approximating architectures on infinite-dimensional spaces [Formula: see text] which, at the same time, have sufficient expressive power to approximate continuous functions on [Formula: see text], are specified by a finite number of parameters only and are “stable” with respect to these parameters.