We study the evolution of hidden-weight spectra in wide neural networks trained by (stochastic) gradient descent. We develop a two-level dynamical mean-field theory (DMFT) that jointly tracks bulk and outlier spectral dynamics for spiked ensembles whose spike directions remain statistically dependent on the random bulk. We apply this framework to two settings: (1) infinite-width nonlinear networks in mean-field/$μ$P scaling and (2) deep linear networks in the proportional high-dimensional limit, where width, input dimension, and sample size diverge with fixed ratios. Our theory predicts how outliers evolve with training time, width, output scale, and initialization variance. In deep linear networks, $μ$P yields width-consistent outlier dynamics and hyperparameter transfer, including width-stable growth of the leading NTK mode toward the edge of stability (EoS). In contrast, NTK parameterization exhibits strongly width-dependent outlier dynamics, despite converging to a stable large-width limit. We show that this bulk+outlier picture is descriptive of simple tasks with small output channels, but that tasks involving large numbers of outputs (ImageNet classification or GPT language modeling) are better described by a restructuring of the spectral bulk. We develop a toy model with extensive output channels that recapitulates this phenomenon and show that edge of the spectrum still converges for sufficiently wide networks.

Launching next week on T-Mobile's network, the cell plan takes a nuclear approach to online safety. A new US-wide cell phone network marketed to Christians is set to launch next week. It blocks porn, which experts in network security say marks the first time a US cell plan has used network-level blocking for such content that can't be turned off even by adult account owners. It's also rolling out a filter on sexual content aimed at blocking material related to gender and trans issues, which will be optional but turned on by default across all plans. The network, which is currently being tested ahead of its May 5 launch date, will be run by Radiant Mobile, a newly launched mobile virtual network operator (MVNO). These operators don't own cell towers but buy bandwidth from the big providers (in this case, T-Mobile) and sell to specific demographics (President Trump announced his own MVNO last year called Trump Mobile; CREDOMobile sends donations to progressive causes).

HTTP/3 is a new application layer protocol supported by most browsers. It uses QUIC as an underlying transport protocol. QUIC provides multiple benefits, like faster connection establishment, reduced latency, and improved connection migration. Hence, popular browsers like Chrome/Chromium, Microsoft Edge, Apple Safari, and Mozilla Firefox have started supporting it. This paper presents an HTTP/3-supported browser dataset collection tool named H3B.

Code is available at https://github.com/zfkarl/DANCE.

In a recent filing, Elon Musk's aerospace company requested to build an orbital data center that relies on solar power. Elon Musk and his aerospace company have requested to build a network that's 100 times the number of satellites that are currently in orbit. On Friday, SpaceX filed an application with the Federal Communications Commission (FCC) to launch a million satellites meant to create an orbital data center. This isn't the first time we're hearing of Musk's plans to build an orbital data center, as it was mentioned by company insiders following the news that the CEO was reportedly preparing to take SpaceX public . According to the filing spotted by, this data center would run off solar power and deliver computing capacity for artificial intelligence needs .

Apple's Siri AI will be powered by Gemini How to claim Verizon's $20 credit for Wednesday's service outage It isn't applied automatically, because of course it isn't. Verizon is offering a very small after Wednesday's massive outage, which drew more than 1.5 million reports on Downdetector and lasted hours. The carrier posted on X that it will offer a $20 credit, but customers must redeem it in the myVerizon app. This credit isn't meant to make up for what happened. No credit really can, the company wrote.

A major Verizon outage appeared to impact customers across the United States starting around noon ET on Wednesday. Calls to Verizon customers from other carriers may also be impacted. Customers of the telecom giant Verizon began reporting cellular outages around the United States beginning around noon ET on Wednesday, saying they could not complete calls and did not have access to mobile data. Verizon broadband internet customers are also reporting issues. AT&T and T-Mobile customers also began reporting service outages in the same timeframe, however these reports may be linked to the Verizon outage.

Apple's Siri AI will be powered by Gemini Issues appear to be concentrated in the eastern United States. Verizon's network appears to be having technical issues that are impacting calls and wireless data. Users on X have reported seeing "SOS" rather than the traditional network bars on their smartphones, and even Verizon's own network status page is struggling to load. Based on the experience of Verizon users on Engadget's staff, the services that are impacted appear to be calls and wireless data. Text messages continue to be delivered normally.

In this paper, we construct a class of stochastic interpolation neural network operators (SINNOs) with random coefficients activated by sigmoidal functions. We establish their boundedness, interpolation accuracy, and approximation capabilities in the mean square sense, in probability, as well as path-wise within the space of second-order stochastic (random) processes \( L^2(Ω, \mathcal{F},\mathbb{P}) \). Additionally, we provide quantitative error estimates using the modulus of continuity of the processes. These results highlight the effectiveness of SINNOs for approximating stochastic processes with potential applications in COVID-19 case prediction.

Topology identification and inference of processes evolving over graphs arise in timely applications involving brain, transportation, financial, power, as well as social and information networks. This chapter provides an overview of graph topology identification and statistical inference methods for multidimensional relational data. Approaches for undirected links connecting graph nodes are outlined, going all the way from correlation metrics to covariance selection, and revealing ties with smooth signal priors. To account for directional (possibly causal) relations among nodal variables and address the limitations of linear time-invariant models in handling dynamic as well as nonlinear dependencies, a principled framework is surveyed to capture these complexities through judiciously selected kernels from a prescribed dictionary. Generalizations are also described via structural equations and vector autoregressions that can exploit attributes such as low rank, sparsity, acyclicity, and smoothness to model dynamic processes over possibly time-evolving topologies. It is argued that this approach supports both batch and online learning algorithms with convergence rate guarantees, is amenable to tensor (that is, multi-way array) formulations as well as decompositions that are well-suited for multidimensional network data, and can seamlessly leverage high-order statistical information.