Quivering Karmelo Anthony is convicted of murdering Austin Metcalf, 17... but now prosecutors have granted him Hail Mary that could see him jailed for as little as TWO YEARS She's always by Trump's side, trusted with the White House's biggest secrets... and she influences millions Inside Travis Kelce's plan to become'the Shaq of the NFL' after wedding Taylor Swift Leaked transcript of UNAIRED 60 Minutes interview exposes REAL reason'callous' CBS star Scott Pelley'deserved to be fired' Woke Canadian lawmakers fly into hilarious rage after conservative asks country's top scientist to define a woman I watched footage of the race crime that split America. Eva Longoria reunites with ex Tony Parker 15 years after cheating scandal split... as shocked fans react Caitlyn Jenner biographer and Robin Riker's ex William Hasley found dead on hiking trail at 78 My compulsive bathroom habit that so many are guilty of left me in excruciating pain. DR STUART reveals early signs... cures that work in days... and when to worry Epstein's massage fixer looks PETRIFIED as she's dragged into explosive congressional grilling - and reveals jaw-dropping'blackmail' theory Zodiac killer case takes bombshell turn as unsolved cipher is CRACKED... and America's top codebreakers say evidence is all pointing to one man Shamed ex mayor Misty Roberts is sentenced to 90 DAYS as she's branded a'predator with hair extensions' by enraged mother of 17-year-old sex assault victim Trump ERUPTS behind closed doors as top Republican pleads with him to axe Tulsi Gabbard's spy-chief replacement Trump's $70B immigration crackdown passes the House as sneaky loophole allows $1.8B weaponization'slush fund' to survive China's Nostradamus issues chilling warning about Trump's UFO file release: 'Atrocities are coming' A professor dubbed ' China's Nostradamus' has made a chilling prediction after the Trump administration released previously classified UFO files. Jiang Xueqin, a Chinese-Canadian educator and political commentator, earned the nickname after making a series of geopolitical predictions that supporters say later came true. Among them were forecasts that Donald Trump would return to the White House in 2024 and that the United States and Israel would become involved in a conflict with Iran under his administration.

Quivering Karmelo Anthony is convicted of murdering Austin Metcalf, 17... but now prosecutors have granted him Hail Mary that could see him jailed for as little as TWO YEARS Trump's $70B immigration crackdown passes the House as sneaky loophole allows $1.8B weaponization'slush fund' to survive I watched footage of the race crime that split America. She's always by Trump's side, trusted with the White House's biggest secrets... and she influences millions Trump ERUPTS behind closed doors as top Republican pleads with him to axe Tulsi Gabbard's spy-chief replacement Leaked transcript of UNAIRED 60 Minutes interview exposes REAL reason'callous' CBS star Scott Pelley'deserved to be fired' Epstein's massage fixer looks PETRIFIED as she's dragged into explosive congressional grilling - and reveals jaw-dropping'blackmail' theory Caitlyn Jenner biographer and Robin Riker's ex William Hasley found dead on hiking trail at 78 Eva Longoria reunites with ex Tony Parker 15 years after cheating scandal split... as shocked fans react Shamed ex mayor Misty Roberts is sentenced to 90 DAYS as she's branded a'predator with hair extensions' by enraged mother of 17-year-old sex assault victim My compulsive bathroom habit that so many are guilty of left me in excruciating pain. DR STUART reveals early signs... cures that work in days... and when to worry Inside Travis Kelce's plan to become'the Shaq of the NFL' after wedding Taylor Swift Moment Real Housewives star Lenny Hochstein's sexual assault accuser'dances' as she leaves Star Island mansion - before filing $100k civil lawsuit Madonna's wild sex claim about JFK Jr now draws surprising response from his outspoken nephew Jack Schlossberg Zodiac killer case takes bombshell turn as unsolved cipher is CRACKED... and America's top codebreakers say evidence is all pointing to one man Kennedy heir Jack Schlossberg concedes Trump is a'genius' as aspiring congressman reveals what he'deeply respects' about president'Great' mom, 32, tried to gas herself and her three young kids to death after inviting them to'popcorn sleepover' in car, prosecutors allege Want to lose up to a stone in six weeks, plus boost your mood and energy levels? Fermented foods from kefir to kombucha are having a moment, hailed for their gut health benefits. But experts say we could be overlooking one of the healthiest ferments out there: sauerkraut.

Stochastic gradient Langevin dynamics combined with Gibbs updates (SGLD--Gibbs) provides a highly scalable approach to approximate Bayesian inference in latent variable models. However, it remains unclear how to tune the algorithm's hyperparameters in a principled manner to ensure the uncertainty estimates are statistically meaningful. In this work, we address this gap in tuning guidance by developing a statistical scaling limit theory for SGLD--Gibbs. We derive a joint asymptotic limit for the global parameters and latent variables under appropriate space-time rescaling. We show that global parameters converge to a diffusion-type limit, while each latent variable converges to a jump process, reflecting the use of intermittent Gibbs updates. This joint jump-diffusion structure reveals how latent-variable randomness contributes to the stationary distribution of the global parameters. We leverage our results to propose explicit guidance on hyperparameter tuning for SGLD--Gibbs that ensures meaningful uncertainty quantification. Numerical experiments show that SGLD--Gibbs with our tuning guidance leads to better parameter estimates, uncertainty quantification, and predictive performance than stochastic variational inference.

This paper studies how efficiently deep ReLU neural networks can approximate and learn smooth functions. When the error is measured in $L^p([0,1]^d)$ norm and the approximator is a network with width $W$ and depth $L$, recent works have proven the supper approximation rate $\mathcal{O}((WL)^{-2s/d})$ for Besov space $\mathcal{B}^s_{q,r}([0,1]^d)$ under the Sobolev embedding condition $s/d>1/q-1/p$. In order to overcome the curse of dimensionality in this rate, we extent this result to anisotropic and mixed smooth function classes. We establish the approximation rate $\mathcal{O}((WL)^{-2\tilde{s}})$ for anisotropic Besov space $\mathcal{B}^{\boldsymbol{s}}_{q,r}([0,1]^d)$ with anisotropic smoothness $\boldsymbol{s}=(s_1,\dots,s_d)$ under the embedding condition $\tilde{s} > 1/q-1/p$, where the mean smoothness $\tilde{s} = (\sum_{i=1}^d s_i^{-1})^{-1}$. For mixed smooth Besov space $\mathcal{MB}^s_{q,r}([0,1]^d)$ with mixed smoothness $s>1/q-1/p$, we show that the approximation rate $\mathcal{O}((WL)^{-2s})$ holds up to logarithmic factors. Using these results, we also derive approximation bounds for the composition of anisotropic Besov functions. As an application, it is shown that deep ReLU neural networks can achieve minimax optimal rates up to logarithmic factors for a wide range of smooth function classes.

MSI's Claw 8 EX AI+ handheld comes out on June 23 MSI's Claw 8 EX AI+ handheld comes out on June 23 The premium gaming handheld's price is expected to be around $1,500. A new wave of premium handhelds is on the way and MSI's upcoming Claw 8 EX AI+ is leading the charge. The company revealed its latest handheld during a hands-on event at its headquarters in Taipei, Taiwan before Computex 2026. The Claw 8 EX AI+ will follow up MSI's previous handhelds introduced in 2024 that start at around $800, but is reportedly going to cost as much as $1,500 . It comes out on June 23.

Two lawmakers in the United States Congress are pushing to repeal a section in an upcoming defence budget that would deepen integration between the US and Israeli militaries. Their effort comes amid outcry against the latest draft of the annual National Defense Authorization Act (NDAA), a bill that sets the budget for the US military. On Sunday, Democratic Congressman Ro Khanna said he would use his seat on the House Armed Services Committee to put forward an amendment to remove Section 224 from the bill. His Republican colleague Thomas Massie also warned that, if the bill is approved by the committee, he would work to oppose it on the floor of the House of Representatives. He framed the issue as a question of US independence from foreign governments.

Why Iran won't give up Hormuz Could Israel sabotage US-Iran deal? Lawmakers in the United States are quietly advancing a proposal that could deepen military ties between the US and Israel in unprecedented ways, at a time when public support for Israel among Americans is increasingly fractured. Among the provisions included in the 2027 National Defence Authorisation Act (NDAA) released this week is Section 224, the "United States-Israel Defence Technology Cooperation Initiative". Some legislators have already signalled opposition, with Representative Thomas Massie saying he would seek to remove the provision if it reaches the House floor. The measure remains at an early stage, but analysts say if passed, it would limit political oversight over the defence relationship.

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A Japan Ground Self-Defense Force Type-88 surface-to-ship missile is fired during the Balikatan exercises at Culili Point Sand Dunes in Paoay, Ilocos Norte province, Philippines, on May 6. | REUTERS Singapore - Defense Minister Shinjiro Koizumi and his Philippine counterpart, Gilberto Teodoro, affirmed Sunday that talks will be launched on the export of surface-to-ship missiles from Japan to the Southeast Asian nation. Koizumi revealed this in talks with reporters after holding a meeting with the Philippine defense chief in Singapore earlier in the day. Type-88 surface-to-ship guided missiles of Japan's Ground Self-Defense Force are expected to be up for consideration. The Philippine side is believed to have shown an interest in procuring the missiles as the Self-Defense Forces used them in the Balikatan multilateral exercises conducted in Manila between April and May. The SDF, which had taken part in the annual exercises organized by the United States and the Philippines as an observer since 2012, joined the drills on a full scale for the first time this year following the entry into force of the Japan-Philippine reciprocal access agreement in September 2025. The possible procurement of Type-88 missiles is expected to help reinforce the deterrent and response capabilities of the Philippines, which is in a territorial dispute with China in the South China Sea.

Bogota - Colombians vote Sunday in a presidential election that will determine the conflict-ridden nation's response to spiraling violence, either staying left and opting for dialogue or tacking right towards all-out war. The constitution forbids a second term for the country's first-ever leftist President Gustavo Petro, whose "total peace" strategy has failed to negotiate an end to conflict with armed groups. Despite his absence from the ballot, "the campaign revolves around Petro," said Yann Basset, political science professor at Bogota's University of Rosario. In a time of both misinformation and too much information, quality journalism is more crucial than ever. By subscribing, you can help us get the story right. With your current subscription plan you can comment on stories.