Recent years have seen significant progress in developing spiking neural networks (SNNs) as a potential solution to the energy challenges posed by conventional artificial neural networks (ANNs). However, our theoretical understanding of SNNs remains relatively limited compared to the ever-growing body of literature on ANNs. In this paper, we study a discrete-time model of SNNs based on leaky integrate-and-fire (LIF) neurons, referred to as discrete-time LIF-SNNs, a widely used framework that still lacks solid theoretical foundations. We demonstrate that discrete-time LIF-SNNs realize piecewise constant functions defined on polyhedral regions, and more importantly, we quantify the network size required to approximate continuous functions. Moreover, we investigate the impact of latency (number of time steps) and depth (number of layers) on the complexity of the input space partitioning induced by discrete-time LIF-SNNs. Our analysis highlights the importance of latency and contrasts these networks with ANNs that use piecewise linear activation functions. Finally, we present numerical experiments to support our theoretical findings.

The lack of trustworthiness is a major downside of deep learning. To mitigate the associated risks clear obligations of deep learning models have been proposed via regulatory guidelines. Therefore, a crucial question is to what extent trustworthy deep learning can be realized. Establishing trust-worthiness requires that the factors influencing an algorithmic computation can be retraced, i.e., the algorithmic implementation is transparent. Motivated by the observation that the current evolution of deep learning models necessitates a change in computing technology, we derive a mathematical framework that enables us to analyze whether a transparent implementation in a given computing model is feasible. We exemplarily apply our trustworthiness framework to analyze deep learning approaches for inverse problems in digital and analog computing models represented by Turing and Blum-Shub-Smale Machines, respectively. Based on previous results, we find that Blum-Shub-Smale Machines have the potential to establish trustworthy solvers for inverse problems under fairly general conditions, whereas, Turing machines cannot guarantee trustworthiness to the same degree. For a longer version of this paper with more details and proofs, we refer to [1].

Deep neural networks have seen tremendous success over the last years. Since the training is performed on digital hardware, in this paper, we analyze what actually can be computed on current hardware platforms modeled as Turing machines, which would lead to inherent restrictions of deep learning. For this, we focus on the class of inverse problems, which, in particular, encompasses any task to reconstruct data from measurements. We prove that finite-dimensional inverse problems are not Banach-Mazur computable for small relaxation parameters. Even more, our results introduce a lower bound on the accuracy that can be obtained algorithmically.