Luca Galimberti

Luca Galimberti, Anastasis Kratsios, Giulia Livieri · eth-zurich

Several non-linear operators in stochastic analysis, such as solution maps to stochastic differential equations, depend on a temporal structure which is not leveraged by contemporary neural operators designed to approximate general maps between Banach space. This paper therefore proposes an operator learning solution to this open problem by introducing a deep learning model-design framework that takes suitable infinite-dimensional linear metric spaces, e.g. Banach spaces, as inputs and returns a universal \textit{sequential} deep learning model adapted to these linear geometries specialized for the approximation of operators encoding a temporal structure. We call these models \textit{Causal Neural Operators}. Our main result states that the models produced by our framework can uniformly approximate on compact sets and across arbitrarily finite-time horizons Hölder or smooth trace class operators, which causally map sequences between given linear metric spaces. Our analysis uncovers new quantitative relationships on the latent state-space dimension of Causal Neural Operators, which even have new implications for (classical) finite-dimensional Recurrent Neural Networks. In addition, our guarantees for recurrent neural networks are tighter than the available results inherited from feedforward neural networks when approximating dynamical systems between finite-dimensional spaces.

Nils Detering, Luca Galimberti, Anastasis Kratsios et al. · eth-zurich

Since their introduction by Kipf and Welling in $2017$, a primary use of graph convolutional networks is transductive node classification, where missing labels are inferred within a single observed graph and its feature matrix. Despite the widespread use of the network model, the statistical foundations of transductive learning remain limited, as standard inference frameworks typically rely on multiple independent samples rather than a single graph. In this work, we address these gaps by developing new concentration-of-measure tools that leverage the geometric regularities of large graphs via low-dimensional metric embeddings. The emergent regularities are captured using a random graph model; however, the methods remain applicable to deterministic graphs once observed. We establish two principal learning results. The first concerns arbitrary deterministic $k$-vertex graphs, and the second addresses random graphs that share key geometric properties with an Erdős-Rényi graph $\mathbf{G}=\mathbf{G}(k,p)$ in the regime $p \in \mathcal{O}((\log (k)/k)^{1/2})$. The first result serves as the basis for and illuminates the second. We then extend these results to the graph convolutional network setting, where additional challenges arise. Lastly, our learning guarantees remain informative even with a few labelled nodes $N$ and achieve the optimal nonparametric rate $\mathcal{O}(N^{-1/2})$ as $N$ grows.

Luca Galimberti

Leveraging the infinite dimensional neural network architecture we proposed in arXiv:2109.13512v4 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 $\mathfrak X$ 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 $\mathfrak Y$ or a topological vector space $E$. Similarly to arXiv:2109.13512v4, 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 $\mathfrak X$ which, at the same time, have sufficient expressive power to approximate continuous functions on $\mathfrak X$, are specified by a finite number of parameters only and are "stable" with respect to these parameters.

Fred Espen Benth, Nils Detering, Luca Galimberti

We propose a new methodology for pricing options on flow forwards by applying infinite-dimensional neural networks. We recast the pricing problem as an optimization problem in a Hilbert space of real-valued function on the positive real line, which is the state space for the term structure dynamics. This optimization problem is solved by facilitating a novel feedforward neural network architecture designed for approximating continuous functions on the state space. The proposed neural net is built upon the basis of the Hilbert space. We provide an extensive case study that shows excellent numerical efficiency, with superior performance over that of a classical neural net trained on sampling the term structure curves.