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Abstract Signal detection is one of the main challenges in data science. As often happens in data analysis, the signal in the data may be corrupted by noise. There is a wide range of techniques that aim to extract the relevant degrees of freedom from data. However, some problems remain difficult. This is notably the case for signal detection in almost continuous spectra when the signal-to-noise ratio is small enough. This paper follows a recent bibliographic line, which tackles this issue with field-theoretical methods. Previous analysis focused on equilibrium Boltzmann distributions for an effective field representing the degrees of freedom of data. It was possible to establish a relation between signal detection and ${\mathbb{Z}}_2$-symmetry breaking. In this paper, we consider a stochastic field framework inspired by the so-called ‘model A’, and show that the ability to reach, or not reach, an equilibrium state is correlated with the shape of the dataset. In particular, by studying the renormalization group of the model, we show that the weak ergodicity prescription is always broken for signals that are small enough, when the data distribution is close to the Marchenko–Pastur law. This, in particular, enables the definition of a detection threshold in the regime where the signal-to-noise ratio is small enough. 

Abstract We continue earlier efforts in computing the dimensions of tangent space cohomologies of Calabi–Yau manifolds using deep learning. In this paper, we consider the dataset of all Calabi–Yau four-folds constructed as complete intersections in products of projective spaces. Employing neural networks inspired by state-of-the-art computer vision architectures, we improve earlier benchmarks and demonstrate that all four non-trivial Hodge numbers can be learned at the same time using a multi-task architecture. With 30% (80%) training ratio, we reach an accuracy of 100% for $h^{(1,1)}$and 97% for $h^{(2,1)}$(100% for both), 81% (96%) for $h^{(3,1)}$, and 49% (83%) for $h^{(2,2)}$. Assuming that the Euler number is known, as it is easy to compute, and taking into account the linear constraint arising from index computations, we get 100% total accuracy.