Subspace recovery from corrupted and missing data is crucial for various applications in signal processing and information theory. To complete missing values and detect column corruptions, existing robust Matrix Completion (MC) methods mostly concentrate on recovering a low-rank matrix from few corrupted coefficients w.r.t. standard basis, which, however, does not apply to more general basis, e.g., Fourier basis. In this paper, we prove that the range space of an $m\times n$ matrix with rank $r$ can be exactly recovered from few coefficients w.r.t. general basis, though $r$ and the number of corrupted samples are both as high as $O(\min\{m,n\}/\log^3 (m+n))$. Our model covers previous ones as special cases, and robust MC can recover the intrinsic matrix with a higher rank. Moreover, we suggest a universal choice of the regularization parameter, which is $\lambda=1/\sqrt{\log n}$. By our $\ell_{2,1}$ filtering algorithm, which has theoretical guarantees, we can further reduce the computational cost of our model. As an application, we also find that the solutions to extended robust Low-Rank Representation and to our extended robust MC are mutually expressible, so both our theory and algorithm can be applied to the subspace clustering problem with missing values under certain conditions. Experiments verify our theories.

ALISON STEWART, PBS NEWSHOUR WEEKEND ANCHOR: Joining me now via Skype to discuss the significance of the U.S. military strike on the Taliban's leader is Jennifer Glasse, a freelance reporter, now in Afghanistan's capital, Kabul. Jennifer, tell us a little bit more about Mansour, who he was within the hierarchy of the Taliban? JENNIFER GLASSE, FREELANCE REPORTER: He was the Taliban leader who took over in last summer. He was a bit of a controversy because when he took over after the announcement, that Mullah Omar had been dead for more than two years. There was a bit of a power struggle and division among the Taliban.

At the top of the corporate world, Apple and Google are in a back-and-forth battle to be number one. It's not clear which of the two Silicon Valley giants will emerge on top in a contest which highlights the contrast of very different business models. Apple then regained, lost and recovered the leader position in May in a battle that appears set to continue for some time. At the close Friday, Apple was worth some 522 billion, to 496 billion for Alphabet. The two companies have both been hugely profitable in recent years, for different reasons. Apple has delivered a line of must-have iPhones and other gadgets that have set trends around the world but now "appears to be a little bit immobile," says Roger Kay, analyst at Endpoint Technologies Associates.

Pakistan accused the United States on Sunday of violating its sovereignty with a drone strike against the leader of the Afghan Taliban, in perhaps the most high-profile US incursion into Pakistani territory since the 2011 raid to kill Osama bin Laden. Afghanistan said the attack killed Mullah Akhtar Mansoor, which, if confirmed, could trigger a succession battle within the armed group that has proved resilient despite a decade and a half of US military deployments to Afghanistan. Afghanistan's Chief Executive Abdullah Abdullah said on Twitter that he was dead, the country's spy agency also said he had been killed, and a source close to Mansoor told Al Jazeera he believed the reports to be true. The Saturday drone strike, which US officials said was authorised by President Barack Obama, showed the US was prepared to go after the Taliban leadership in Pakistan, which the government in Kabul has repeatedly accused of sheltering the rebels. Pakistan protested on Sunday, saying the US government did not inform Prime Minister Nawaz Sharif beforehand.

China is emerging as the biggest market for video games. It is, therefore, great news to investors of Microsoft (NASDAQ:MSFT) that its subsidiary Mojang has stuck a deal with NetEase (NASDAQ:NTES). Mojang granted a 5-year exclusive license to NetEase to distribute the mobile and PC versions of its cult hit game, Minecraft. Mojang will create a China-specific version of Minecraft for NetEase. Microsoft paid 2.5 billion to acquire Mojang in 2014.

The killing of Afghan Taliban leader Mullah Mohammed Akhtar Mansour in a U.S. drone strike was greeted Sunday by Kabul's political leadership as a game-changer in efforts to end the long insurgent war plaguing Afghanistan. In a rare show of unity, President Ashraf Ghani and Chief Executive Abdullah Abdullah both welcomed the news of Mansour's death as the removal of a man who unleashed violence against innocent civilians in Afghanistan and was widely regarded as an obstacle to peace within the militant group. Mansour, believed to be in his 50s, was killed when a U.S. drone fired on his vehicle in the southwestern Pakistan province of Baluchistan, although there were conflicting accounts whether the airstrike occurred Friday or Saturday. He had emerged as the successor to Taliban founder Mullah Mohammad Omar, whose 2013 death was only revealed last summer. Mansour "engaged in deception, concealment of facts, drug-smuggling and terrorism while intimidating, maiming and killing innocent Afghans," Ghani said in a statement on his official Twitter account.

The manufacturing hub for the electronics industry, Kunshan, in Jiangsu province, is seeking a drastic reduction in labour costs as it undergoes a makeover after an industrial explosion killed 146 people in 2014. The county, one-seventh the size of neighbouring Shanghai and the mainland's first county to achieve US 4,000 per capita income, was adjudged the best county for its economic performance by Forbes for seven years in a row. However, the blaze, blamed on poor safety standards and haphazard industrialisation, dented Kunshan's pride. More than a year on, the county, which attracts much of its investment from Taiwan, is trying to reinvent its growth strategy. It is accelerating growth by replacing humans with robots and encouraging start-ups.

Tech journals and TV segments are overflowing with them. But bots have been around for a very long time. Microsoft's relentlessly chirpy paperclip assistant "Clippy" seems to have morphed into the PR train wreck of Tay. The global titans of social technology are bullish about a brave new Artificial Intelligence (AI)-powered world of enhanced apps. Last month Microsoft CEO Satya Nadella proclaimed that "bots are the new apps" during the company's Build developers conference.

Can artificial intelligence lay paint to paper to create meaning and evoke emotion? The answer lies in the results from the first Robot Art competition. The contest, organized by artist Andrew Conru who holds a Ph.D. in mechanical engineering design from Stanford University, aims to foster innovation in A.I. and integrate aesthetics and technology. It was open to students from universities and high schools and awarded cash prizes in two categories: fully automated execution or manually generated. Here are the results from the 10 winners, which were chosen thanks to feedback from the public as well as that from a panel of judges.

Afghan Taliban leader Mullah Akhtar Mansour has been killed, Afghanistan's National Directorate of Security confirmed Sunday afternoon after several hours of uncertainty. The Afghan intelligence agency said Mansour, who was officially named the group's leader last year, was killed in an "airstrike" in a remote area in Balochistan in southwestern Pakistan Saturday. Prior to that, Afghan President Ashraf Ghani also confirmed that Mansour had been targeted in a drone strike by the United States. "Mullah Akhtar Mansour refused to answer repeated calls by the people and Government of Afghanistan to end the war and violence in the country. While sheltering himself in hideouts outside Afghanistan, he was also involved in deception, concealment of facts, maiming and killing innocent Afghans, terrorism, intimidation, drug smuggling as well as obstruction of development and progress in Afghanistan as he obstinately insisted on continuing the war," the Afghan president's office said in a statement Sunday.