Wide neural networks trained with weight decay provably exhibit neural collapse

Deep neural networks (DNNs) at convergence consistently represent the training data in the last layer via a geometric structure referred to as neural collapse. This empirical evidence has spurred a line of theoretical research aimed at proving the emergence of neural collapse, mostly focusing on the unconstrained features model. Here, the features of the penultimate layer are free variables, which makes the model data-agnostic and puts into question its ability to capture DNN training. Our work addresses the issue, moving away from unconstrained features and studying DNNs that end with at least two linear layers. We first prove generic guarantees on neural collapse that assume (i) low training error and balancedness of linear layers (for within-class variability collapse), and (ii) bounded conditioning of the features before the linear part (for orthogonality of class-means, and their alignment with weight matrices). The balancedness refers to the fact that W⊤ℓ+1Wℓ+1 ≈ WℓW⊤ℓfor any pair of consecutive weight matrices of the linear part, and the bounded conditioning requires a well-behaved ratio between largest and smallest non-zero singular values of the features. We then show that such assumptions hold for gradient descent training with weight decay: (i) for networks with a wide first layer, we prove low training error and balancedness, and (ii) for solutions that are either nearly optimal or stable under large learning rates, we additionally prove the bounded conditioning. Taken together, our results are the first to show neural collapse in the end-to-end training of DNNs

PMLR

A standard model that arises in several applications in sequential decision-making is partially observable Markov decision processes (POMDPs) where a decision-making agent interacts with an uncertain environment. A basic objective in POMDPs is the reachability objective, where given a target set of states, the goal is to eventually arrive at one of them. The limit-sure problem asks whether reachability can be ensured with probability arbitrarily close to 1. In general, the limit-sure reachability problem for POMDPs is undecidable. However, in many practical cases, the most relevant question is the existence of policies with a small amount of memory. In this work, we study the limit-sure reachability problem for POMDPs with a fixed amount of memory. We establish that the computational complexity of the problem is NP-complete

Multivariate Gini-type discrepancies

Measuring distances in a multidimensional setting is a challenging problem, which appears in many fields of science and engineering. In this paper, to measure the distance between two multivariate distributions, we introduce a new measure of discrepancy which is scale invariant and which, in the case of two independent copies of the same distribution, and after normalization, coincides with the scaling invariant multidimensional version of the Gini index recently proposed in [P. Giudici, E. Raffinetti and G. Toscani, Measuring multidimensional inequality: A new proposal based on the Fourier transform, preprint (2024), arXiv:2401.14012 ]. A byproduct of the analysis is an easy-to-handle discrepancy metric, obtained by application of the theory to a pair of Gaussian multidimensional densities. The obtained metric does improve the standard metrics, based on the mean squared error, as it is scale invariant. The importance of this theoretical finding is illustrated by means of a real problem that concerns measuring the importance of Environmental, Social and Governance factors for the growth of small and medium enterprises

The PLATO mission

PLATO (PLAnetary Transits and Oscillations of stars) is ESA’s M3 mission designed to detect and characterise extrasolar planets and perform asteroseismic monitoring of a large number of stars. PLATO will detect small planets (down to <2R Earth) around bright stars (<11 mag), including terrestrial planets in the habitable zone of solar-like stars. With the complement of radial velocity observations from the ground, planets will be characterised for their radius, mass, and age with high accuracy (5%, 10%, 10% for an Earth-Sun combination respectively). PLATO will provide us with a large-scale catalogue of well-characterised small planets up to intermediate orbital periods, relevant for a meaningful comparison to planet formation theories and to better understand planet evolution. It will make possible comparative exoplanetology to place our Solar System planets in a broader context. In parallel, PLATO will study (host) stars using asteroseismology, allowing us to determine the stellar properties with high accuracy, substantially enhancing our knowledge of stellar structure and evolution. The payload instrument consists of 26 cameras with 12cm aperture each. For at least four years, the mission will perform high-precision photometric measurements. Here we review the science objectives, present PLATO‘s target samples and fields, provide an overview of expected core science performance as well as a description of the instrument and the mission profile towards the end of the serial production of the flight cameras. PLATO is scheduled for a launch date end 2026. This overview therefore provides a summary of the mission to the community in preparation of the upcoming operational phases

Universal behavior of the BCS energy gap

We consider the BCS energy gap „.T / (essentially given by „.T / .T; p/, the BCS order parameter) at all temperatures 0 T Tc up to the critical one, Tc, and show that, in the limit of weak coupling, the ratio „.T /=Tc is given by a universal function of the relative temperature T =Tc. On the one hand, this recovers a recent result by Langmann and Triola [Phys. Rev. B 108 (2023), no. 10, article no. 104503] on three-dimensional s-wave superconductors for temperatures bounded uniformly away from Tc. On the other hand, our result lifts these restrictions, as we consider arbitrary spatial dimensions d 2 ¹1; 2; 3º, discuss superconductors with non-zero angular momentum (primarily in two dimensions), and treat the perhaps physically most interesting (due to the occurrence of the superconducting phase transition) regime of temperatures close to Tc. ​

The convergence of heavy and light seeds to overmassive black holes at cosmic dawn

The James Webb Space Telescope has revealed low-luminosity active galactic nuclei at redshifts of z ≳ 4–7, many of which host accreting massive black holes (BHs) with BH-to-galaxy mass (MBH/M⋆) ratios exceeding the local values by more than an order of magnitude. The origin of these overmassive BHs remains unclear but requires potential contributions from heavy seeds and/or episodes of super-Eddington accretion. We present a growth model coupled with dark matter halo assembly to explore the evolution of the MBH/M⋆ ratio under different seeding and feedback scenarios. Given the gas inflow rates in protogalaxies, BHs grow episodically at moderate super-Eddington rates, and the mass ratio increases early on, despite significant mass loss through feedback. Regardless of seeding mechanisms, the mass ratio converges to a universal value ∼0.1–0.3, set by the balance between gas feeding and star formation efficiency in the nucleus. This behavior defines an attractor in the MBH–M⋆ diagram, where overmassive BHs grow more slowly than their hosts, while undermassive seeds experience rapid growth before aligning with the attractor. We derive an analytical expression for the universal mass ratio, linking it to feedback strength and halo growth. The convergence of evolutionary tracks erases seeding information from the mass ratio by z ∼ 4–6. Detecting BHs with ∼105−6 M⊙ at higher redshifts that deviate from the convergence trend would provide key diagnostics of their birth conditions

Non–Hermitian spectral universality at critical points

For general large non–Hermitian random matrices X and deterministic normal deformations A, we prove that the local eigenvalue statistics of A + X close to the critical edge points of its spectrum are universal. This concludes the proof of the third and last remaining typical universality class for non–Hermitian random matrices (for normal deformations), after bulk and sharp edge universalities have been established in recent years

Discovery of two new polars evolved past the period bounce

We report the discovery of two new magnetic cataclysmic variables with brown dwarf companions and long orbital periods (P_{\rm orb}=95\pm1 and 104\pm2 min). This discovery increases the sample of candidate magnetic period bouncers with confirmed sub-stellar donors from four to six. We also find their X-ray luminosity from archival XMM–Newton observations to be in the range L_{\rm X}\approx10^{28}-10^{29} \mathrm{erg\,s^{-1}} in the 0.25–10 keV band. This low luminosity is comparable with the other candidates, and at least an order of magnitude lower than the X-ray luminosities typically measured in cataclysmic variables. The X-ray fluxes imply mass transfer rates that are much lower than predicted by evolutionary models, even if some of the discrepancy is due to the accretion energy being emitted in other bands, such as via cyclotron emission at infrared wavelengths. Although it is possible that some or all of these systems formed directly as binaries containing a brown dwarf, it is likely that the donor used to be a low-mass star and that the systems followed the evolutionary track for cataclysmic variables, evolving past the period bounce. The donor in long period systems is expected to be a low-mass, cold brown dwarf. This hypothesis is supported by near-infrared photometric observations that constrain the donors in the two systems to be brown dwarfs cooler than 1100 K (spectral types T5 or later), most likely losing mass via Roche Lobe overflow or winds. The serendipitous discovery of two magnetic period bouncers in the small footprint of the XMM–Newton catalogue implies a large space density of these type of systems, possibly compatible with the prediction of 40–70 per cent of magnetic cataclysmic variables to be period bouncers

STIM1-induced widening of non-pore-lining TM interfaces is crucial for Orai1 pore opening

The Ca2+-release-activated Ca2+ (CRAC) channel Orai1 is activated by interaction with the Ca2+ sensor Stromal Interaction Molecule 1 (STIM1). Owing to the lack of structurally resolved Orai1/STIM1 complexes, the impact of their coupling on individual Orai1 transmembrane domain (TM) movements is unclear. This study investigates STIM1-independent and STIM1-dependent Orai1-TM dynamics using photocrosslinking unnatural amino acids (UAAs) at each individual TM position. We primarily identify CRAC-channel-like currents directly after UAA incorporation or additional UV-light irradiation at TM3 sites that interface with non-pore-lining TMs. Using UAAs combined with conventional site-directed mutagenesis and molecular dynamics simulations, we discover that pore opening involves a widening of interfaces formed by TM3 with non-pore-lining TMs. Orai1 mutants with a UAA in TM3 exhibit weaker STIM1-induced activation after UV exposure, possibly caused by a restricted widening of non-pore-lining TM interfaces. We demonstrate that photocrosslinking UAAs are excellent tools for improving our understanding of key determinants and ion channel dynamics modulating pore opening

The ionizing photon production efficiency of star-forming galaxies at z ∼ 4–10

Context. Investigating the ionizing emission of star-forming galaxies and the escape fraction of ionizing photons is critical to understanding their contribution to reionization and their impact on the surrounding environment. The number of ionizing photons available to reionize the intergalactic medium (IGM) depends on not only the abundance of galaxies but also their efficiency in producing ionizing photons (ξion). This quantity is thus fundamental to quantify the role of faint versus bright sources in driving this process, as we must assess their relative contribution to the total ionizing emissivity. Aims. Our goal is to estimate the ξion using Balmer lines (Hα or Hβ) in a sample of 761 galaxies at 4 ≤ z ≤ 10 selected from different JWST spectroscopic surveys. We aim to determine the redshift evolution of ξion and the relation of ξion with the physical properties of the galaxies. Methods. We used the available HST and JWST photometry to perform a spectral energy distribution (SED) fitting in the sample to determine their physical properties and relate them with ξion. We used the BAGPIPES code for the SED fitting and assumed a delayed exponential model for the star formation history. We used the NIRSpec spectra from prism or grating configurations to estimate Balmer luminosities, and then constrained ξion values after dust correction. Results. We find a mean value of 1025.22 Hz erg−1 for ξion in the sample with an observed scatter of 0.42 dex. We find an increase in the median values of ξion with redshift from 1025.09 Hz erg−1 at z ∼ 4.18 to 1025.28 Hz erg−1 at z ∼ 7.14, confirming the redshift evolution of ξion found in other studies. Regarding the relation between ξion and physical properties, we find a decrease in ξion with increasing stellar mass, indicating that low-mass galaxies are efficient producers of ionizing photons. We also find an increase in ξion with increasing specific star formation rate (sSFR) and increasing UV absolute magnitude. This indicates that faint galaxies and galaxies with high sSFR are also efficient producers. We also investigated the relation of ξion with the rest-frame equivalent width (EW) of [OIII]λ5007 and find that galaxies with the higher EW([OIII]λ5007) are more efficient producers of ionizing photons, with the best fit leading to the relation log(ξion)  =  0.43 × log(EW[OIII])+23.99. Similarly, we find that galaxies with higher O32 = [OIII]λ5007/[OII]λλ3727,3729 and lower gas-phase metallicities (based on the R23 = ([OIII]λλ4959,5007+[OII]λλ3727,3729)/Hβ calibration) show higher ξion values