Blind inverse problems arise in many experimental settings where the forward operator is partially or entirely unknown. In this context, methods developed for the non-blind case cannot be adapted in a straightforward manner. Recently, data-driven approaches have been proposed to address blind inverse problems, demonstrating strong empirical performance and adaptability. However, these methods often lack interpretability and are not supported by rigorous theoretical guarantees, limiting their reliability in applied domains such as imaging inverse problems. In this work, we shed light on learning in blind inverse problems within the simplified yet insightful framework of Linear Minimum Mean Square Estimators (LMMSEs). We provide an in-depth theoretical analysis, deriving closed-form expressions for optimal estimators and extending classical results. In particular, we establish equivalences with suitably chosen Tikhonov-regularized formulations, where the regularization depends explicitly on the distributions of the unknown signal, the noise, and the random forward operators. We also prove convergence results under appropriate source condition assumptions. Furthermore, we derive rigorous finite-sample error bounds that characterize the performance of learned estimators as a function of the noise level, problem conditioning, and number of available samples. These bounds explicitly quantify the impact of operator randomness and reveal the associated convergence rates as this randomness vanishes. Finally, we validate our theoretical findings through illustrative numerical experiments that confirm the predicted convergence behavior.

BBC Verify has been tracking the Marinera for weeks. Housing, Europe ties, economy... what Canadians are hopeful for in 2026 The BBC spoke to people in Toronto and Montreal to find out what they're optimistic about heading into the new year. The powerful storm system brought blizzard conditions to areas of the Midwest and East Coast causing some travel delays. Governor Gavin Newsom has declared a state of emergency for parts of California, including Los Angeles, San Bernardino and San Diego. The White Settlement Police Department is searching for two suspects.

Website of Claude seen in an iPhone screen on 21 May 2023. Website of Claude seen in an iPhone screen on 21 May 2023. Anthropic is planning a $10bn fundraise that would value the Claude chatbot maker at $350bn, according to multiple reports published on Wednesday. The new valuation represents an increase of nearly double from about four months ago, per CNBC, which reported that the company had signed a term sheet that stipulated the $350bn figure. The round could close within weeks, although the size and terms could change.

Grok Is Generating Sexual Content Far More Graphic Than What's on X A WIRED review of outputs hosted on Grok's official website shows it's being used to create violent sexual images and videos, as well as content that includes apparent minors. Elon Musk's Grok chatbot has drawn outrage and calls for investigation after being used to flood X with "undressed" images of women and sexualized images of what appear to be minors. However, that's not the only way people have been using the AI to generate sexualized images. Grok's website and app, which are are separate from X, include sophisticated video generation that is not available on X and is being used to produce extremely graphic, sometimes violent, sexual imagery of adults that is vastly more explicit than images created by Grok on X. It may also have been used to create sexualized videos of apparent minors.

NASA astronaut Nichole Ayers spotted the summer storm while aboard the International Space Station. Breakthroughs, discoveries, and DIY tips sent every weekday. Lightning is one of Earth's most impressive phenomena. The sudden discharges of superheated plasma occur even in seemingly sunny conditions, rip apart air molecules, and can easily span hundreds of miles . But while there is still a lot to learn about lightning from our perspective here on Earth, there's also much to glean by observing it from high above.

Breakthroughs, discoveries, and DIY tips sent every weekday. Critics have long warned that too much television rots your brain, but new evidence suggests at least time in front of the tube may actually benefit your mental health . In a study published in the, researchers combined their own investigations into leisure time with information from the United States Census Bureau on household size, stress levels, and screen usage. People who take some "Me Time" at home after work appear to have an easier time bouncing back to their daily responsibilities. "Household size is really about how many demands a person experiences when they go home," Soo Min Toh, a behaviorist at the University of Toronto Mississauga and study co-author, said in a university profile .

New Year is a time of reinvention. In that spirit, I would like to shake up this preview of 2026's best sci-fi and science-related TV with a radical act: including a series that started last year. That may seem strange, but the second season of Fallout (Amazon Prime Video) aired in only mid-December, so, for my money, it counts. Set in a retrofuturistic US, generations of humans have lived inside radiation-proof bunkers sold to them by the shadowy Vault-Tec corporation. Last season, former vault-dweller Lucy (Ella Purnell) went surface-side to find her missing father, encountering cowboys and cannibals along the way.

Sarah Owen, chair of the committee, said'we do not view it as appropriate to use such a platform to share our work'. Sarah Owen, chair of the committee, said'we do not view it as appropriate to use such a platform to share our work'. The influential Commons women and equalities committee has decided to stop using X after the social media site's AI tool began generating thousands of digitally altered images of women and children with their clothes removed. The move by the cross-party committee places renewed pressure on ministers to take decisive action after the site was flooded with images including sexualised and unclothed pictures of children, generated by its AI tool, Grok. Sarah Owen, the Labour MP who chairs the committee, said that given preventing violence against women and girls was among its key policy areas, "it has become increasingly clear that X is not an appropriate platform to be using for our communications".

Plus: Amazon has listed retailers' goods without their permission Last spring, 3,000 British soldiers deployed an invisible automated intelligence network, known as a "digital targeting web," as part of a NATO exercise called Hedgehog in the damp forests of Estonia's eastern territories. The system had been cobbled together over the course of four months--an astonishing pace for weapons development, which is usually measured in years. Its purpose is to connect everything that looks for targets--"sensors," in military lingo--and everything that fires on them ("shooters") to a single, shared wireless electronic brain. Eighty years after total war last transformed the continent, the Hedgehog tests signal a brutal new calculus of European defense. But leaning too much on this new mathematics of warfare could be a risky bet. This story is from the next print issue of magazine.

Things to Do in L.A. Tim Walz isn't the only governor plagued by fraud. Minnesota Gov. Tim Walz said he would not seek a third term amid attacks over a fraud scandal involving child care funding. This is read by an automated voice. Please report any issues or inconsistencies here . California has lost billions to cheats in the last few years, leaving Newsom vulnerable to the same sort of attack that took down Walz.