The reasons of why local iterative algorithms like gradient descent are able to converge towards good solutions in high-dimensional and non-convex landscapes currently remain a complete mystery in several cases. In this work, we focus on a single-layer neural network with a quadratic activation function, a reminiscence of a common problem called phase retrieval. Leveraging methods and intuitions from statistical physics, we show the success of gradient descent is conditioned by a transition in the Hessian of some peculiar states in the loss landscapes stopping the dynamics when the dimension is infinite. When it is finite, we show this picture changes drastically and the system exploits the initial information to slip away from the bad minima and find back the good solution before reaching the bad ones.

Generative models, with at its forefront diffusion models, are becoming central in both science and industry with many relevant applications. We however still lack a theoretical understanding of how they are able to create new samples. In this work, we provide the first quantitative analytical analysis of the behaviour of the generative backward dynamics using methods from statistical physics.

This paper exposes the reachable accuracies of the redshift-space constraints provided by the several environments of the cosmic web with respect to the matter monopole and quadrupole statistics. By splitting the density field into its cosmic web components, one can tighten down the constraints by factors up to 5.5 for the summed neutrino mass.

This work aims at extracting the cosmological information content of the several cosmic web environments. While we know that the matter power spectrum is not containing all the information about hte underlying cosmological model, we can wonder wether the environments are enclosing different types of information that one can use to break some of the degeneracies among parameters of the model. In particular, we show that a simple two-point correlator becomes sensitive to higher-order features when we have a look at the environments instead of the full matter distribution.

The extraction of patterns from spatially structured point-cloud datasets is ubiquitous in many fields of science. In this work, we address the case of extracting one-dimensional structure from such data by formulating the problems in terms of a regularised version of a mixture model.