This paper builds a hierarchy of explicit, non-asymptotic tail bounds for the supremum of the Kostlan--Shub--Smale (KSS) random field on the sphere, and applies it to two problems: Spiked Tensor PCA and the landscape of the spherical $k$-spin model. For Tensor PCA, we study the non-asymptotic statistical limits of estimating a rank-$R$ symmetric signal tensor of order~$k\ge 3$ and dimension~$d\ge 3$ from a single Gaussian observation at signal-to-noise ratio~$λ$, through the \emph{profile maximum likelihood estimator}, the MLE restricted to normalized rank-$R$ tensors of coherence at least~$κ$. Our analysis uses a single reduction: a deterministic geometric inequality (the Tube Method) and a rank-reduction step bound the estimation error by the supremum of the canonical KSS field, which the Kac--Rice formula turns into a Gaussian integral against the expected absolute characteristic polynomial of a shifted Gaussian Orthogonal Ensemble, controlled in turn by the four explicit tail bounds of our hierarchy (three from a Mehta--Fyodorov representation, one from a Ben Arous--Dembo--Guionnet large deviation). The same reduction yields two results, each with explicit constants. For estimation, a finite-$(k,d)$ error bound recovers the asymptotically optimal rate~$\sqrt{d\log k}$ of Perry, Wein and Bandeira, with explicit dependence on the rank~$R$ and the coherence~$κ$. For the landscape, a two-sided non-asymptotic bracketing of the annealed complexity of the spherical $k$-spin Hamiltonian recovers the Auffinger--Ben Arous--Černý complexity function in the high-dimensional limit.

We study sparse semi-continuous tensor data with excess zeros, heavy right tails, and slice-specific dispersion. Such features arise naturally in monetary-valued multi-way data, such as international trade, where most exporter--importer--product--year cells are zero while positive values are continuous and highly variable. To model these data, we propose a Bayesian hierarchical tensor factorization model that places a low-rank CP structure on a latent Poisson rate tensor and couples it with a conditional Gamma model for positive outcomes, with rate parameters that can vary across slices within a mode. The model therefore separates the occurrence and magnitude of positive observations while borrowing strength across all tensor dimensions through a shared low-rank latent structure. To scale posterior inference to large arrays, we develop a hybrid variational--Monte Carlo algorithm that combines efficient coordinate ascent updates with a partially collapsed augmented-data sampler. Applied to approximately 60 million trade flows, the method surfaces multiway dependence across exporters, importers, products, and years that is difficult to recover from gravity-type or pairwise network analyses, which do not jointly model the product and temporal dimensions.

A botanist at Kew's Madagascar research site scans a plant for digitisation. A botanist at Kew's Madagascar research site scans a plant for digitisation. AI could help win'race against extinction' of vital plants, say botanists Tech is helping to identify and save new specimens and could open'genomic goldmine' of fungi data The rise of AI and digitisation could be a turning point in the "race against extinction" faced by botanists trying to identify and save vital plants before they vanish, according to a major report from Royal Botanic Gardens, Kew. New technology is enabling scientists to track how flowering times have shifted by weeks around the world, rapidly identify new specimens and even get crucial genetic data from 180-year-old fungus specimens, potentially opening a "genomic goldmine". Digitisation and online access to millions of specimens that were until now only accessible in archives is also producing new insights, especially in the global south.

We propose a Koopman-enhanced physics-informed neural network (K--PINN) framework for parameter inference and forecasting in nonlinear epidemic models. This method combines Koopman operator theory and physics-informed learning. It maps epidemic states into a latent observable space where the dynamics evolve approximately linearly while satisfying the governing epidemic equations through automatic differentiation. This integration improves interpretability, parameter identifiability, and long-term predictive stability. We apply the proposed framework to a normalized SEIRSD epidemic model and evaluate it using synthetic monkeypox (Mpox) data and real-world datasets from Germany, Morocco, and Sweden for the SARS-CoV-2 virus. Synthetic trajectories are generated using a structure-preserving, nonstandard finite difference scheme to ensure reliable training data. Numerical results demonstrate that K--PINN achieves more accurate parameter estimation, trajectory reconstruction, and long-term forecasting than classical PINNs and Koopman-EDMD approaches. These results suggest that K--PINN is an effective machine learning framework for epidemic modeling that can be extended to more complex systems.

Large Language Models (LLMs) require significant GPU memory when processing long texts, with the key value (KV) cache consuming up to 70% of total memory during inference. Although existing compression methods reduce memory by evaluating the importance of individual tokens, they overlook critical semantic relationships between tokens, resulting in fragmented context and degraded performance. We introduce ChunkKV, which fundamentally reimagines KV cache compression by treating semantic chunks - rather than isolated tokens - as basic compression units. This approach preserves complete linguistic structures and contextual integrity, ensuring that essential meaning is retained even under aggressive compression. Our innovation includes a novel layer-wise index reuse technique that exploits the higher cross-layer similarity of preserved indices in ChunkKV, reducing computational overhead and improving throughput by 26.5%. Comprehensive evaluations on challenging benchmarks: LongBench, Needle-InA-HayStack, GSM8K, and JailbreakV demonstrate that ChunkKV outperforms state-of-the-art methods by up to 8.7% in precision while maintaining the same compression ratio. These results confirm that semantic-aware compression significantly enhances both efficiency and performance for long-context LLM inference, providing a simple yet effective solution to the memory bottleneck problem. The code is available at link.

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].