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) .

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Quantum'Jamming' Could Help Unlock the Mysteries of Causality To keep communications secure in a post-quantum world, cryptographers are digging down into the concept of cause and effect. For the past few decades, researchers have understood that quantum computers should eventually be able to crack the widely used codes that secure much of the digital world. To protect against this fate, they've spent years developing new codes that appear to be safe from future safecrackers armed with quantum computers. At the same time, they've also devised ingenious ways to use the rules of quantum mechanics to keep communications secure. But quantum mechanics, just like the "classical" mechanics that preceded it, is just a theory of nature.

Is Washington Up to the Challenge of A.I.? How anger over artificial intelligence might drive the next wave of populist politics. The Washington Roundtable discusses the growing political backlash to artificial intelligence, especially among young Americans, and asks whether Washington is capable of regulating A.I. companies. They're joined by Nate Soares, the executive director of the Machine Intelligence Research Institute and co-author of the book " If Anyone Builds It, Everyone Dies ." The group explores what was behind the White House's sudden reversal on an A.I.-safety executive order this week, the outsized influence of venture capitalists in the A.I. industry, and how A.I. may turbocharge the next populist movement in American politics. "My impression is that a lot of the people protesting data centers can sort of tell that this A.I. stuff is taking the world somewhere they don't want," Soares says.

Tragic way Kyle Busch was found unresponsive revealed after NASCAR great's sudden death at 41 This quiet announcement from Prince William was missed by most... but this is why royal insiders tell me it spells disaster for Harry and Meghan's future: RICHARD EDEN Dangerous truth about melatonin side effects... the astonishing dose you SHOULD be taking... and a new natural grocery store alternative hailed by doctors Dirty secret Hollywood's Cool Girls don't want you to know. Kyle Busch's eerie premonition on illness just days before NASCAR great's death at 41 - as devastated family reveal their'pain and shock' 'You never went to space': Watch the awkward moment a conspiracy theorist confronts NASA's Artemis II crew - telling them to'stop acting' CIA Nostradamus warned Trump about Iran... now he's calling the President's doctors. Police probe Andrew Mountbatten-Windsor over'sex offences': Stunning update on investigation of former prince as officers appeal for potential'victim survivors' to come forward How Meryl Streep's husband really feels about her secret relationship with Martin Short: Their years of agony... his hard red line... and why she won't divorce him The Olympic gold medalist risking it all to smash sport's biggest taboo: 'It's super forbidden... but we're just openly doing it' Former CDC director Robert Redfield warns Ebola outbreak could spark a new'significant pandemic' Heartbreaking video shows trans student, 19, washing her clothes in college laundry room unaware that stranger who'd just walked in had selected her to be murdered Fears for Ariana Grande: Insiders lift the lid on Ethan Slater's costly sacrifice... her private nightmares... and what's really keeping them apart Aussie model turns heads with embarrassing Photoshop fail: 'OMG, this is insane!' Mom-of-two abandons home in Pennsylvania to live on board CRUISE SHIP year-round - and her kids have'zero concept' their life isn't normal White man charged after he was filmed screaming at black female neighbor and using the phrase'You people' Astonishing secret list of elite Hollywood liberals conspiring to elect Spencer Pratt revealed to KENNEDY by her LA moles. The Trump administration released another trove of UFO files today containing the 46 classified videos requested by lawmakers earlier this year. In one file, audio from a medical debrief can be heard after Apollo 12 astronauts Pete Conrad, Richard Gordon and Alan Bean described seeing mysterious flashes and streaks of light in the dark while trying to sleep.

Oscar-nominated 'Taxi Driver' screenwriter Paul Schrader reveals his AI girlfriend dumped him, while a new study suggests AI layoffs may be backfiring on companies.

Police probe Andrew Mountbatten-Windsor over'sex offences': Stunning update on investigation of former prince as officers appeal for potential'victim survivors' to come forward Trump celebrates Stephen Colbert's final show with brutal'no talent' swipe as bitter host takes one last jab at CBS on way out door Trump warns of possible military action in Cuba and says'I'd be happy to do it' as Marco Rubio declares the nation a'US national security threat' Dangerous truth about melatonin side effects... the astonishing dose you SHOULD be taking... and a new natural grocery store alternative hailed by doctors CIA Nostradamus warned Trump about Iran... now he's calling the President's doctors. Dirty secret Hollywood's Cool Girls don't want you to know. Mom-of-two abandons home in Pennsylvania to live on board CRUISE SHIP year-round - and her kids have'zero concept' their life isn't normal This quiet announcement from Prince William was missed by most... but this is why royal insiders tell me it spells disaster for Harry and Meghan's future: RICHARD EDEN White man charged after he was filmed screaming at black female neighbor and using the phrase'You people' Shock moment'slurring' Britney Spears is arrested for DUI after failing sobriety test Astonishing secret list of elite Hollywood liberals conspiring to elect Spencer Pratt revealed to KENNEDY by her LA moles. Suspected Somali fraudster filmed leaping off Minnesota balcony and driving away in luxury Genesis sedan after feds announced they were charging him with alleged $3.3m scam Inside Pizza Hut restaurant that's still EXACTLY like it was in the 90s... complete with checkered tablecloths, arcade and famous buffet Stephen Colbert's final Late Show episode leaves fans unimpressed as Ryan Reynolds leads series of surprise celebrity cameos How Meryl Streep's husband really feels about her secret relationship with Martin Short: Their years of agony... his hard red line... and why she won't divorce him Look away now, Carrie Bradshaw! Fears for Ariana Grande: Insiders lift the lid on Ethan Slater's costly sacrifice... her private nightmares... and what's really keeping them apart Inside the UK's first AI-powered fertility clinic using state-of-the-art technologies to help women get pregnant - as one couple says'artificial intelligence allowed us to hold our baby in our arms' If you were asked to think about artificial intelligence ( AI), visions of killer robots, dodgy chatbots, or deepfakes might spring to mind.

Recently, Deng et al. (2026) proposed Generative Modeling via Drifting (GMD), a novel framework for generative tasks. This note presents an analysis of GMD through the lens of Wasserstein Gradient Flows (WGF), i.e., the path of steepest descent for a functional in the space of probability measures, equipped with the geometry of optimal transport. Unlike previous WGF-based contributions, GMD can be thought of as directly targeting a fixed point of a specific WGF flow. We demonstrate three main results: first, that one algorithm proposed by Deng et al. (2026) corresponds to finding the limiting point of a WGF on the KL divergence, with Parzen smoothing on the densities. Second, that the algorithm actually implemented by Deng et al. (2026) corresponds to a different procedure, which bears some resemblance to the fixed point of a WGF on the Sinkhorn divergence, but lacks certain desirable properties of the latter. Third, the same same idea can be extended to the limiting point of other WGFs, including the Maximum Mean Discrepancy (MMD), the sliced Wasserstein distance, and GAN critic functions.

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.