Former Olympian is arrested for allegedly vandalizing Reflecting Pool... but he claims he merely touched it Call me cynical, but the real reason Gruesome Twosome Harry and Meghan are returning to the UK is just so obvious... and highly humiliating: MAUREEN CALLAHAN Three more arrested over bungee jumper's death after she was hurled from bridge without a rope I lost 50lb without jabs using this easy but overlooked method. But I still felt dowdy - until I discovered these expert anti-ageing fashion and beauty tips. Inside America's new fattest town: Burgers are the size of your head, gyms lie empty and custom mobility scooters carry 800lb loads... as we investigate why Ozempic just DOESN'T work Blake Lively runs errands in frumpy outfit after reconciling with ex-BFF Taylor Swift... miles away from reported'bachelorette party' I've spoken to thousands of children who claim they can recall a past life ... these chilling stories have convinced me they're telling the truth Stingy fast food giant named America's favorite restaurant AGAIN... and experts think they know why Ex-partner of dad who was berated for taking his daughters into women's bathroom claims he'exploited' girls and accuses him of failing to pay child support... before he hits back The'marry me' sex move that'll make even the most commitment-phobic of men beg to see you again... and it worked for THREE of my friends America's next real estate time bomb detonates the sun-kissed southern housing dream: 'New condo crisis' sparks chilling warning as it snakes across the nation Furious Trump hits back at Italian Prime Minister Meloni and gives her unusual'nickname' as their photo feud ramps up TV star mom, 46, who appeared on'quitting everything to change your life' show died in fire at luxury Caribbean beach resort that sent 1,700 tourists running for their lives Grace Kelly's lookalike granddaughter, 27, wows in bikini snaps...as she packs on the PDA during beach getaway The four mistakes that led to bungee tragedy on Skeleton Bridge: FRED KELLY saw the scene for himself, now he retraces the prelude to disaster. So was it really an accident? Famous TV mansion left standing after Malibu's harshest wildfires struggled to find a buyer for 14 years but finally sells for an eye-watering price Swedish actress, 81, was in TWO James Bond movies and also worked with Charlton Heston, who is she?

More than 1,000 civilians in Sudan have been killed in drone strikes in the first five months of 2026, according to the United Nations. The death toll is due to a "sharp" increase in the use of drone warfare in the country's vicious civil war, UN High Commissioner for Human Rights (UNHCHR) Volker Turk said in a speech on Monday. On top of documenting more than 1,000 civilians being killed in the first five months of this year, the UN office also reported "rampant" levels of sexual violence, including rape. The war in the African nation started in April 2023 when a rivalry between Sudan's army chief, Abdel Fattah al-Burhan, and the commander of the paramilitary Rapid Support Forces, Mohamed Hamdan "Hemedti" Dagalo, exploded into war. The conflict, which had first started in the capital Khartoum, soon spread to several areas of the country.

Former Olympian seen in handcuffs as Trump threatens'years in jail' and more arrests after vandals SABOTAGE Reflecting Pool with'corrosive and destructive chemicals' Angelina Jolie's son Pax, 22, surfaces in LA after bombshell revelation about his relationship to Brad Pitt Keir Starmer'will announce as early as Monday that he is quitting as Prime Minister' after spending weekend locked in tense talks about his future with his wife Victoria at Chequers Mortifying truth about Clavicular's'botched' nose job: Infertile influencer's'trans' admission to friends... as insider reveals what's said behind closed doors - and twisted secrets that'll leave fans floored Giorgia Meloni rips'senseless' attacks from Trump as Italian Prime Minister refuses to back down amid G7 feud Inside America's new fattest town: Burgers are the size of your head, gyms lie empty and custom mobility scooters carry 800lb loads... as we investigate why Ozempic just DOESN'T work Call me cynical, but the real reason Gruesome Twosome Harry and Meghan are returning to the UK is just so obvious... and highly humiliating: MAUREEN CALLAHAN Stingy fast food giant named America's favorite restaurant AGAIN... and experts think they know why I lost 50lb without jabs using this easy but overlooked method. But I still felt dowdy - until I discovered these expert anti-ageing fashion and beauty tips. No one can see the real reason Jelly Roll divorced Bunnie XO. Blake Lively runs errands in frumpy outfit after reconciling with ex-BFF Taylor Swift... miles away from reported'bachelorette party' Embattled Alexi Lalas makes controversial World Cup declaration amid tension with Fox colleagues: 'Makes you look like a weak poser' Scientists propose radical new theory of consciousness - and claim it doesn't depend on flesh and blood Candace Owens hits out at nasty rumors claiming she was DEAD... as fellow MAGA influencer claims her account was hacked Grace Kelly's lookalike granddaughter, 27, wows in bikini snaps...as she packs on the PDA during beach getaway TV star mom, 46, who appeared on'quitting everything to change your life' show died in fire at luxury Caribbean beach resort that sent 1,700 tourists running for their lives The four mistakes that led to bungee tragedy on Skeleton Bridge: FRED KELLY saw the scene for himself, now he retraces the prelude to disaster. So was it really an accident?

Former Olympian seen in handcuffs as Trump threatens'years in jail' and more arrests after vandals SABOTAGE Reflecting Pool with'corrosive and destructive chemicals' Angelina Jolie's son Pax, 22, surfaces in LA after bombshell revelation about his relationship to Brad Pitt Mortifying truth about Clavicular's'botched' nose job: Infertile influencer's'trans' admission to friends... as insider reveals what's said behind closed doors - and twisted secrets that'll leave fans floored Keir Starmer'will announce as early as Monday that he is quitting as Prime Minister' after spending weekend locked in tense talks about his future with his wife Victoria at Chequers Inside America's new fattest town: Burgers are the size of your head, gyms lie empty and custom mobility scooters carry 800lb loads... as we investigate why Ozempic just DOESN'T work Call me cynical, but the real reason Gruesome Twosome Harry and Meghan are returning to the UK is just so obvious... and highly humiliating: MAUREEN CALLAHAN I lost 50lb without jabs using this easy but overlooked method. But I still felt dowdy - until I discovered these expert anti-ageing fashion and beauty tips. Giorgia Meloni rips'senseless' attacks from Trump as Italian Prime Minister refuses to back down amid G7 feud No one can see the real reason Jelly Roll divorced Bunnie XO. Wyndham Clark's stunning girlfriend pays tribute to polarizing golfer as he stands on the brink of US Open glory TV star mom, 46, who appeared on'quitting everything to change your life' show died in fire at luxury Caribbean beach resort that sent 1,700 tourists running for their lives Embattled Alexi Lalas makes controversial World Cup declaration amid tension with Fox colleagues: 'Makes you look like a weak poser' Scientists propose radical new theory of consciousness - and claim it doesn't depend on flesh and blood Stingy fast food giant named America's favorite restaurant AGAIN... and experts think they know why Blake Lively runs errands in frumpy outfit after reconciling with ex-BFF Taylor Swift... miles away from reported'bachelorette party' Grace Kelly's lookalike granddaughter, 27, wows in bikini snaps...as she packs on the PDA during beach getaway The animal with a face'not even a mother would love': Elusive goblin shark is seen alive in its natural habitat for the first time A goblin shark has been seen alive in its natural habitat - not just once, but twice. Scientists were reviewing footage captured at a seamount near Jarvis Island in 2019 when they spotted the elusive animal.

We study the dynamics of stochastic gradient descent (SGD) for a class of sequence models termed Sequence Single-Index (SSI) models, where the target depends on a single direction in input space applied to a sequence of tokens. This setting generalizes classical single-index models to the sequential domain, encompassing simplified one-layer attention architectures. We derive a closed-form expression for the population loss in terms of a pair of sufficient statistics capturing semantic and positional alignment, and characterize the induced high-dimensional SGD dynamics for these coordinates. Our analysis reveals two distinct training phases: escape from uninformative initialization and alignment with the target subspace, and demonstrates how the sequence length and positional encoding influence convergence speed and learning trajectories. These results provide a rigorous and interpretable foundation for understanding how sequential structure in data can be beneficial for learning with attention-based models. Stochastic Gradient Descent (SGD) is the core optimization tool driving modern machine learning. Recent years have seen substantial progress in understanding its dynamics, particularly in two-layer networks [Saad and Solla, 1995, Mei et al., 2018, Chizat and Bach, 2018, Rotskoff and VandenEijnden, 2022, Sirignano and Spiliopoulos, 2020, Arnaboldi et al., 2023a]. While global convergence is qualitatively well-understood when the network is wide enough, quantitative results are scarcer. A particularly fruitful body of recent theoretical work addressing this gap has focused on deriving precise convergence rates for particular model classes on synthetic data, such as high-dimensional Gaussian single and multi-index models [Ben Arous et al., 2021, Abbe et al., 2022, 2023].

Former Olympian seen in handcuffs as Trump threatens'years in jail' and more arrests after vandals SABOTAGE Reflecting Pool with'corrosive and destructive chemicals' Angelina Jolie's son Pax, 22, surfaces in LA after bombshell revelation about his relationship to Brad Pitt Mortifying truth about Clavicular's'botched' nose job: Infertile influencer's'trans' admission to friends... as insider reveals what's said behind closed doors - and twisted secrets that'll leave fans floored Keir Starmer'will announce as early as Monday that he is quitting as Prime Minister' after spending weekend locked in tense talks about his future with his wife Victoria at Chequers Inside America's new fattest town: Burgers are the size of your head, gyms lie empty and custom mobility scooters carry 800lb loads... as we investigate why Ozempic just DOESN'T work Call me cynical, but the real reason Gruesome Twosome Harry and Meghan are returning to the UK is just so obvious... and highly humiliating: MAUREEN CALLAHAN I lost 50lb without jabs using this easy but overlooked method. But I still felt dowdy - until I discovered these expert anti-ageing fashion and beauty tips. No one can see the real reason Jelly Roll divorced Bunnie XO. Wyndham Clark's stunning girlfriend pays tribute to polarizing golfer as he stands on the brink of US Open glory TV star mom, 46, who appeared on'quitting everything to change your life' show died in fire at luxury Caribbean beach resort that sent 1,700 tourists running for their lives Giorgia Meloni rips'senseless' attacks from Trump as Italian Prime Minister refuses to back down amid G7 feud Scientists propose radical new theory of consciousness - and claim it doesn't depend on flesh and blood Embattled Alexi Lalas makes controversial World Cup declaration amid tension with Fox colleagues: 'Makes you look like a weak poser' Stingy fast food giant named America's favorite restaurant AGAIN... and experts think they know why Blake Lively runs errands in frumpy outfit after reconciling with ex-BFF Taylor Swift... miles away from reported'bachelorette party' Grace Kelly's lookalike granddaughter, 27, wows in bikini snaps...as she packs on the PDA during beach getaway Is this the key to preventing a Super El Niño? As scientists warn that the coming Super El Niño could be the worst in recorded history, one group of researchers has proposed a drastic solution.

We investigate fingerprints in pretraining datasets for large language models (LLMs) through dataset classification experiments. Building on prior work demonstrating the existence of fingerprints or biases in popular computer vision datasets, we analyze popular open-source pretraining datasets for LLMs derived from CommonCrawl including C4, RefinedWeb, DolmaCC, RedPajama-V2, FineWeb, and DCLM-Baseline. Despite those datasets being obtained with similar curation steps, neural networks can classify surprisingly well which dataset a single text sequence belongs to, significantly better than a human can. This indicates that small differences in filtering and processing pipelines induce fingerprints, that we find are evident in formatting, vocabulary, and content distributions. Such fingerprints can negatively impact cross-dataset generalization. Additionally, we show that these fingerprints propagate through training: sequences generated by models trained on those datasets can be accurately classified by a classifier trained on the original datasets. This can offer insights into data characteristics that are typically undisclosed by LLM developers, including pretraining mixture proportions and finetuning data sources.

Future link prediction on temporal graphs is a fundamental task with wide applicability in real-world dynamic systems. These scenarios often involve both recurring (seen) and novel (unseen) interactions, requiring models to generalize effectively across both types of edges. However, existing methods typically rely on complex memory and aggregation modules, yet struggle to handle unseen edges. In this paper, we revisit the architecture of existing temporal graph models and identify two essential but overlooked modeling requirements for future link prediction: representing nodes with unique identifiers and performing target-aware matching between source and destination nodes. To this end, we propose Cross-Attention based Future Link Predictor on Temporal Graphs (CRAFT), a simple yet effective architecture that discards memory and aggregation modules and instead builds on two components: learnable node embeddings and cross-attention between the destination and the source's recent interactions. This design provides strong expressive power and enables target-aware modeling of the compatibility between candidate destinations and the source's interaction patterns. Extensive experiments on diverse datasets demonstrate that CRAFT consistently achieves superior performance with high efficiency, making it well-suited for large-scale real-world applications.

Concertgoer, 51, who plunged to his death in front of horrified wife at Madison Square Garden is identified as'much-loved' dad-of-two Jennifer Lopez enjoys concert night with Ben Affleck's child Fin and her own child Oskar CNN star Jake Tapper slammed for choice of guests for his Father's Day TV special: 'What the heck?' Call me cynical, but the real reason Gruesome Twosome Harry and Meghan are returning to the UK is just so obvious... and highly humiliating: MAUREEN CALLAHAN No one can see the real reason Jelly Roll divorced Bunnie XO. Family-man facade of award-winning children's swim coach is shattered by disturbing teen babysitter claims: Read all the vile texts How to boost your testosterone WITHOUT supplements or risky treatments: Jason, 56, doubled his levels with these simple lifestyle tweaks - and doctors say any man can do the same. Here's how to reap the benefits to your body AND sex life My secret sex fantasy is destroying my marriage. I'm repulsed by my husband... but can't bear to admit what I REALLY want: DEAR JANE Trump sparks confusion after sharing Father's Day photo of'mystery' woman while appearing to call her a'great daughter' Karoline Leavitt flaunts her postpartum body seven weeks after giving birth... and shares gushing tribute to husband, 60, for Father's Day I had sex with my brother.

We consider the problem of how many samples from a Gaussian multi-index model are required to weakly reconstruct the relevant index subspace. Despite its increasing popularity as a testbed for investigating the computational complexity of neural networks, results beyond the single-index setting remain elusive. In this work, we introduce spectral algorithms based on the linearization of a message passing scheme tailored to this problem. Our main contribution is to show that the proposed methods achieve the optimal reconstruction threshold. Leveraging a high-dimensional characterization of the algorithms, we show that above the critical threshold the leading eigenvector correlates with the relevant index subspace, a phenomenon reminiscent of the Baik-Ben Arous-Peche (BBP) transition in spiked models arising in random matrix theory.