We study the fundamental problem of clustering $n$ points into $K$ groups drawn from a mixture of isotropic Gaussians in $\mathbb{R}^d$. Specifically, we investigate the requisite minimal distance $Δ$ between mean vectors to partially recover the underlying partition. While the minimax-optimal threshold for $Δ$ is well-established, a significant gap exists between this information-theoretic limit and the performance of known polynomial-time procedures. Although this gap was recently characterized in the high-dimensional regime ($n \leq dK$), it remains largely unexplored in the moderate-dimensional regime ($n \geq dK$). In this manuscript, we address this regime by establishing a new low-degree polynomial lower bound for the moderate-dimensional case when $d \geq K$. We show that while the difficulty of clustering for $n \leq dK$ is primarily driven by dimension reduction and spectral methods, the moderate-dimensional regime involves more delicate phenomena leading to a "non-parametric rate". We provide a novel non-spectral algorithm matching this rate, shedding new light on the computational limits of the clustering problem in moderate dimension.

Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Revealed: Unexplained objects that stop and accelerate quickly in space detected by'highly qualified observers, says former UFO chief. 'Spacecraft we know don't behave that way' Sign up for our US Editor's Picks newsletter to get all the best exclusive stories The Pentagon's UFO office former chief has revealed unexplained objects were detected in space - and that some performed maneuvers defying anything in America's known aerospace arsenal. Lieutenant Colonel Tim Phillips, who was acting director of the All-domain Anomaly Resolution Office (AARO) until last April, told the Daily Mail that while most cases involved objects in the air, some detections extended beyond the atmosphere.

ROTC students at Old Dominion subdued and killed ISIS-linked gunman who left one dead, two wounded after shouting'Allahu Akbar' and opened fire Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' A 3.5-magnitude earthquake shook central Kansas on Wednesday, jolting an area layered over ancient volcanic rock deep below the surface. The tremor was detected at 2.14pm local time (3.14pm Although Kansas has no active volcanoes, the region marks the southern reach of the Midcontinent Rift System, a massive tectonic event that nearly split North America apart in Earth's distant past.

Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Emperor penguins are on the pathway to EXTINCTION: Satellite images reveal how shrinking sea ice is forcing birds into crowded groups - with potentially'catastrophic' consequences READ MORE: Antarctica's worst-case climate scenario laid bare Emperor penguins are one of the Antarctic's most iconic animals - but these majestic birds are on the pathway to extinction. For the first time, satellite images have captured the penguins' elusive moulting colonies, where they replace their feathers with new waterproof plumage. Moulting is a particularly dangerous time for emperor penguins as they cannot enter the water to feed for several weeks while their new plumage regrows.

Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Earth's'Doomsday Vault' has been updated: Scientists add 7,800 samples into the secretive bunker in the Arctic A bunker hidden in the Arctic might sound like somewhere a Bond villain would hide out - but the real-life'Doomsday Vault' actually serves a very serious purpose. The Svalbard Global Seed Vault, located beneath a mountain in Spitsbergen, a remote island in the Norwegian archipelago of Svalbard, houses'spare copies' of seeds from around the world. That collection has now been updated with over 7,800 seed varieties, including staple grains from Africa, olives from Spain, and ancient crops from Guatemala.

ROTC students at Old Dominion subdued and killed ISIS-linked gunman who left one dead, two wounded after shouting'Allahu Akbar' and opened fire Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Ey up AI! Scientists are teaching artificial intelligence how to understand UK accents and dialects - so, how many of these regional terms do YOU know? Scientists are teaching AI how to understand slang words such as'chuck' and'nowt' so that automated phone lines can understand local callers. Councils across the UK are turning to artificial intelligence to man their phones, helping people with enquiries such as when to put their bins out or how to pay their council tax.

Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Alexander brothers' alleged HIGH SCHOOL gang rape video: Classmates speak out on sick'taking turns' footage... as creepy unseen photos are exposed Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Live Nation executives mocked'stupid' concert-goers in emails where they bragged about how to best rip them off: '$60 for closer grass' NFL superstar Xavier Worthy spills all on Travis Kelce, the Chiefs' struggles... and having Taylor Swift as his No 1 fan Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Nancy Mace throws herself into Iran warzone as she goes rogue on Middle East rescue mission: 'I AM that person' Hidden toxins in kids' treats EXPOSED: Health guru Jillian Michaels' sit-down with Casey DeSantis reveals dangers lurking in popular foods Why are animals picking on Punch the monkey? Scientists have revealed the tragic truth about Punch the monkey - the viral macaque who has stolen the hearts of millions across social media. The seven-month-old Japanese macaque was born at Ichikawa Zoo last year, where he was rejected by his mother. Zookeepers gave him a stuffed orangutan toy, who he quickly formed a bond with - with viral footage showing him clinging to the plushie. Fans were briefly relieved when footage emerged of another macaque grooming and comforting Punch.

Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Alexander brothers' alleged HIGH SCHOOL gang rape video: Classmates speak out on sick'taking turns' footage... as creepy unseen photos are exposed Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting NFL superstar Xavier Worthy spills all on Travis Kelce, the Chiefs' struggles... and having Taylor Swift as his No 1 fan Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Nancy Mace throws herself into Iran warzone as she goes rogue on Middle East rescue mission: 'I AM that person' Hidden toxins in kids' treats EXPOSED: Health guru Jillian Michaels' sit-down with Casey DeSantis reveals dangers lurking in popular foods Declassified CIA files reveal chilling blueprint to manipulate Americans' minds through covert drugging with vaccines READ MORE: CIA mind-control project'programmed Trump shooter', congressman claims A newly released CIA document reveals a chilling blueprint to manipulate minds through covert drugging experiments . The report, added to the CIA's reading room in 2025, details the government's once top-secret Project Artichoke that ran from 1951 to 1956, focusing on behavior control, interrogation techniques and psychological manipulation. The seven-page document, titled'Special Research for Artichoke,' with an attachment labeled'Suggested Fields for Special Research Relative Artichoke,' outlines proposals to develop chemicals capable of altering human behavior. It discusses drugs designed for both immediate effects, like truth serums and long-term influence, potentially administered through food, water, alcohol or cigarettes. Researchers also suggested that such substances could be disguised in medical treatments such as vaccinations or injections.

ROTC students at Old Dominion subdued and killed ISIS-linked gunman who left one dead, two wounded after shouting'Allahu Akbar' and opened fire Horrifying next twist in the Alexander brothers case: MAUREEN CALLAHAN exposes an unthinkable perversion that's been hiding in plain sight Kentucky mother and daughter turn down $26.5MILLION to sell their farms to secretive tech giant that wants to build data center there Hollywood icon who starred in Psycho after Hitchcock dubbed her'my new Grace Kelly' looks incredible at 95 Kylie Jenner's total humiliation in Hollywood: Derogatory rumor leaves her boyfriend's peers'laughing at her' behind her back Tucker Carlson erupts at Trump adviser as she hurls'SLANDER' claim linking him to synagogue shooting Ben Affleck'scores $600m deal' with Netflix to sell his AI film start-up Long hair over 45 is ageing and try-hard. I've finally cut mine off. Alexander brothers' alleged HIGH SCHOOL rape video: Classmates speak out on sickening footage... as creepy unseen photos are exposed Heartbreaking video shows very elderly DoorDash driver shuffle down customer's driveway with coffee order because he is too poor to retire Amber Valletta, 52, was a '90s Vogue model who made movies with Sandra Bullock and Kate Hudson, see her now Model Cindy Crawford, 60, mocked for her'out of touch' morning routine: 'Nothing about this is normal' You think this is bad? The rain in the UK seems to have been never-ending - with some regions experiencing showers every single day this year. Now scientists have warned that things are going to get even worse.

Britain is to develop new air defence weapons alongside the EU's four biggest military powers, deepening ties with the European defence sector. The project will invite manufacturers in the UK, Germany, France, Italy and Poland to submit plans to build low-cost missiles and autonomous drones. The allies are pledging a speedy process to build the weapons together, inspired by Ukraine's development of cheap drones to counter attacks from Russia. The UK's Ministry of Defence (MoD) says the programme will prioritise a lightweight, affordable surface-to-air weapon, with the first project to be delivered by next year. The plan, announced at a meeting of the five countries' defence ministers in the Polish city of Krakow, marks a boost to UK-Europe ties after the failure of talks last year over UK participation in the EU's new €150bn (£130bn) defence fund.