M-Fibration Theory with Applications to Neural Network Compression

The purpose of this paper is to provide a general, comprehensive, theoretical framework that allows one to deal with fibrations on graphs labelled on a commutative monoid. This is a genuine extension of the theory of graph fibrations (as introduced in "Fibrations of Graphs" [Discrete Math., vol. 243, pp. 21-66, 2002]), that makes it possible to deal with weighted graphs, and also graphs labelled with other algebraic structures. The derived theory also lends itself naturally to consider approximate fibrations. As an example, we show how this framework can be applied to the compression of arbitrary neural networks (including CNNs), providing a strong theoretical underpinning to the recent results in "The role of fibration symmetries in geometric deep learning" [Proc. Natl. Acad. Sci. USA, vol. 123, no. 4, p. e2416552123, 2026]

On the Applicability of Safety Nets: A Safety-By-Design Solution for Certifying Neural Networks

The integration of Artificial Intelligence (AI) in safety-critical aviation systems presents significant challenges for certification and deployment. Aviation, often regarded as the safest form of transportation, relies on numerous safety-critical systems. For future safety-critical AI-based systems, EASA requires a Safety-by-Design approach, which can be achieved by using Safety Nets that combine neural network compression with lookup tables to ensure 100 % correct runtime behavior across the discretized operational design domain. Although Safety Nets have been studied, no comprehensive study of their performance characteristics and system design trade-offs has been conducted. This work presents the first systematic analysis of the trade-off between neural network and lookup table size in Safety Nets. By systematically comparing neural networks with diverse architectures, this study identifies optimal design parameters that minimize overall storage and memory requirements while maintaining certification compliance. Results demonstrate that architectures with 3 to 5 hidden layers, each with approximately 50 to 100 nodes, combined with one-hot encoding, achieve the best balance. In these configurations, neural networks accurately represent at least 97 % of the data, while compact lookup tables handle the remaining errors. The resulting Safety Nets reduce the system size by almost three orders of magnitude, fitting within the memory budget of current avionics hardware while guaranteeing 100 % correct outputs across the entire discretized input space, as required by EASA guidelines. This work provides the first-ever open-source implementation of Safety Nets for HCAS and VCAS with replicable results, demonstrating a practical pathway toward certifiable AI-based systems in aviation and establishing Safety Nets as a viable Safety-by-Design solution for safety-critical applications.

Understanding Quantization-Aware Training: Gradients at Quantized Weights Bias to the Low-Loss Basin

Post-training quantization (PTQ) converts a trained full-precision model into low-bit weights without task-level retraining, while quantization-aware training (QAT) incorporates quantization into the training loop. Although PTQ is efficient and often accurate at moderate bitwidths, it can fail sharply at aggressive bitwidths; QAT is more expensive but can often recover the lost accuracy. We propose a unified geometric framework that explains both PTQ failure and QAT recovery. We model full-precision training as following a low-loss \emph{river} inside a wider \emph{valley}: a normal neighborhood of the river forms a nearly flat \emph{basin}, while leaving this basin incurs a sharp loss increase. When the quantization grid is comparable to the basin width, local PTQ objectives, including rounding and Hessian-based second-order reconstruction, can select a high-loss deployed quantized point outside the basin even when nearby low-loss quantized points exist. In this regime, straight-through-estimator-based QAT has a useful bias: it evaluates gradients at the deployed quantized weights while updating latent full-precision weights, causing the gradient to sense the valley wall and acquire an inward component that steers subsequent quantized iterates back into the basin. We formalize this mechanism through a local landscape model, construct a geometric PTQ failure mode, and prove finite-time QAT recovery under local quantizer-compatibility assumptions. Experiments across vision and language models under multiple neural-network quantization schemes corroborate the predicted basin-crossing failure of PTQ and the corresponding recovery mechanism of QAT.