Bradley-Terry-Luce (BTL) model estimation is a well-established strategy to rank a collection of items given a dataset of pairwise comparisons. Although the theoretical performance of BTL estimation methods, such as spectral and maximum likelihood estimation, is well studied in the regime of uniformly sampled graphs, generalizing such results to a wider class of random graphs has proved challenging. In this work, we investigate the entry-wise error of spectral algorithms against a semi-random adversary that can arbitrarily boost the sampling probabilities of certain edges. We find that the performance of the unweighted spectral method is heavily dependent on the spectral properties of the generated graph. Furthermore, we show that asymptotic performance approaching that of uniformly sampled graphs can be recovered by appropriately reweighting the observed edges to counteract the adversary and restore the spectral gap. Finally, we provide numerical simulations that support our theoretical findings.

We develop a gradient flow on the space of probability measures defined on matrix-valued parameters induced by regularized Muon, an analytically smoothed version of the idealized Muon optimizer. The key observation is that the regularized orthogonalization map is the gradient of a smooth Fenchel-dual smoothing of the nuclear norm. This identifies the (regularized) Muon update as a mirror/prox step in the update variable, with momentum acting as the dual coordinate. We use this structure to lift Muon from a single matrix parameter to finite-particle probability objectives of the form $J(ρ)=R\left(\int F d ρ\right)$, a setting motivated by mean-field descriptions of neural-network training, and derive the inertial continuous-time limit. Using this structure, we derive the finite-particle continuous-time limit under the inertial scaling of step size and momentum, and then pass to a phase-space mean-field equation over probability laws on parameter-momentum pairs. The resulting flow can be shown to be a damped Hamiltonian probability dynamics whose kinetic energy is induced by the regularized Muon mirror potential. We prove an exact Hamiltonian dissipation identity, showing that the Hamiltonian energy decreases monotonically. While the target objective itself need not be monotone along the inertial Muon dynamics, under additional gradient-dominance, bounded-momentum, and curvature/alignment assumptions, we obtain continuous and discrete-time exponential convergence rates for the objective gap. We also study the well-posedness of the mean-field limit equation and establish propagation of chaos guarantees for the interacting particle system. Finally, we extend the formulation to Hilbert-valued feature maps on product matrix spaces, yielding a blockwise Muon probability flow applicable to smooth transformer mixture-of-experts models.

We introduce training-free looped transformers, in which a lightweight inference-time wrapper loops a contiguous mid-stack block of layers of a frozen checkpoint without additional fine-tuning, continued training, or architectural changes. Unlike prior looped transformer methods that train with the looped structure end-to-end, we retrofit recurrence onto pretrained models at test time. We show that naive block reapplication usually degrades performance, highlighting the importance of the loop application strategy. Motivated by viewing a pre-norm transformer block as a forward Euler step on an ODE, we instead treat looping as a refinement of the same approximation, replacing one large update with smaller damped sub-steps. Across seven dense, sparse MoE, and MLA+MoE model families, our method improves Qwen3-4B-Instruct by +2.64 pp on MMLU-Pro, Qwen3-30B-A3B-Instruct by +1.14 pp on CommonsenseQA, and Moonlight-16B-A3B-Instruct by +1.20 pp on OpenBookQA.

We consider the problem of sampling from an unnormalized Boltzmann/ Gibbs density, π(θ) exp V(θ),θ Θ Rd, where the normalization constant is unknown (and/or intractable) and only the potential function V (and typically its derivatives) can be evaluated. This problem arises across various domains in Bayesian inference, statistical physics, and modern machine learning. A common variational perspective on sampling is to characterize the target distribution π as the unique minimizer of a functional (typically a divergence functional) over the space of probability measures. From this viewpoint, sampling can be formulated as evolving an initial distribution ρ0 toward π via the gradient flow of this functional under a suitable geometric structure on the space of probability measures. In this paper, we focus on a gradient flow based sampling methodology built from the spherical Hellinger Kantorovich (SHK), also known as the Wasserstein Fisher Rao (WFR), geometry on the space of probability measures (Kondratyev and Vorotnikov, 2019; Liero et al., 2018; Chizat et al., 2015). When the variational objective is the exclusive KL divergence ρ 7 KL(ρ π), the SHK gradient flow generates a time-indexed family of marginals {ρt}t 0 (initialized at ρ0 P2(Θ)) that evolves according to the continuity reaction equation (4). This evolution is equivalent to the birth-death Langevin dynamics introduced in Lu et al. (2019) .

Facilities can be branded as aligned with Scotland's climate goals despite significant emissions, said APRS. Facilities can be branded as aligned with Scotland's climate goals despite significant emissions, said APRS. Scotland's'green datacentres' policy ignores emissions impact of AI, analysis shows A Scottish government policy designed to encourage datacentres to build in Scotland could lead to a massive volume of carbon emissions being ignored, according to an analysis by a Scottish charity. "Green datacentres" are at the heart of Scotland's ambitions to develop economically. Enshrined in national policy, they are part of a larger, UK-wide effort to attract big AI investment to Scotland.

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Figure AI says three humanoid robots named Bob, Frank and Gary ran autonomously for over 24 hours, sorting packages without human control in a warehouse test.

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