In the matrix completion problem, we seek to recover or estimate a matrix, when only a fraction of its entries are observed. While it is impossible to complete an arbitrary matrix using only partial observations of its entries, it may be possible to fully recover matrix entries when the matrix has an appropriate underlying structure. For example, most low-rank matrices can be completed accurately with high probability, by solving a convex optimization problem (Candés and Recht, 2009). Consequently, algorithms for lowrank matrix completion have enjoyed widespread use across many disciplines, including collaborative filtering and recommender systems (Koren et al., 2009), multi-task learning and classification (Amit et al., 2007; Argyriou et al., 2007; Wu and Lange, 2015), computer vision (Chen and Suter, 2004), statistical genetics (Chi et al., 2013), as well as remote sensing (Malek-Mohammadi et al., 2014). In this paper, we consider matrix completion under a structural assumption that is closely related to the low-rank assumption; i.e., we assume that the matrix entries vary "smoothly" with respect to a graphical organization of the rows and columns. For example, in the context of a movie recommendation system, we seek to complete a user-by-movies ratings matrix. We may have additional information about users, such as if pairs of users are friends on a social media application, as well as additional information from a movie database, such as the co-occurrence of certain film principles. We expect the entries of a movie ratings matrix to vary "smoothly" over a neighborhood of users, defined by a friendship graph, and over a neighborhood of movies, defined by a shared movie principles graph. When such local similarity structure exists, and is available, it behooves us to leverage this information to predict missing entries in a matrix.

Amazon may soon be adding'internet service provider' to its long list of skills. People familiar with the matter revealed the eCommerce giant is considering this service, first for its European customers. This new venture would let Amazon bundle internet access with its Prime video streaming, which would ultimately increase its membership base. People familiar with the matter revealed the eCommerce giant is considering this service for its European customers. A person briefed on the discussion told The Information that Amazon is considering offering internet access to its European customers.

The first reviews of the new handset hit the internet today, and while positive, the consensus is that Google's phone is good - but not great. At a special event in San Francisco the firm unveiled a raft of new products - even though most of them have already leaked online. They included two new phones, a Pixel and the 5.5-inch Pixel XL, which go on sale on Thursday. Google achieves that by pulling together the best features from Apple, Samsung and other phones and offering them at prices comparable to iPhones -- starting at about 650 for the regular, 5-inch model and 770 for the 5.5-inch'XL' edition. Both versions go on sale Thursday through Verizon, Best Buy and Google's online store.

Updated 10-18-16: We've updated our recommendation for best Phablet (5.5 inches or greater) to the excellent Pixel XL. We haven't yet reviewed the standard-size Pixel, but one of the two is a likely candidate for best overall phone. The Android universe is teeming with options, from super-expensive flagship phones, to affordable models that make a few calculated compromises, to models expressly designed for, say, great photography. Chances are that whichever phone you buy, you'll keep it for at least two years. So choosing the best Android phone for you isn't a decision you should take lightly.

SAN FRANCISCO - As handy as all our voice recognition friends are, conversing with them still feels you're talking to a foreign relative. Whether its Siri (Apple) or Alexa (Amazon) or Google Assistant or Cortana (Microsoft), each requires the human to speak in slow, articulated phrases to increase the odds of comprehension. But researchers at Microsoft say they've reached a milestone that promises a future where machines can transcribe us as well as another person. In a paper published Monday called "Achieving Human Parity in Conversational Speech Recognition," engineers with Microsoft Artificial Intelligence and Research announced they'd developed a speech recognition system that makes the same or fewer errors as professional transcriptionists. The team hit a word error rate of 5.9 percent, down from the 6.3 percent WER the team reported just last month.

ORLANDO, Fla.--Microsoft CEO Satya Nadella faced sharp questions from Gartner analysts Tuesday about the privacy-invading implications of its 26.2 billion acquisition of LinkedIn, and its all-knowing virtual assistant, Cortana. Helen Huntley, one of the Gartner analysts questioning Nadella at a conference here, was particularly pointed about the fears. Cortana, said Huntley, "knows everything about me when I'm working. She knows what files I'm looking at, she knows what I'm downloading, she knows when I'm working, when I'm not working," she said. Cortana is "big brother intersected ... with productivity," said Huntley.

Why is it that every time humans develop a really clever computer system in the movies, it seems intent on killing every last one of us at its first opportunity? In Stanley Kubrick's masterpiece, 2001: A Space Odyssey, HAL 9000 starts off as an attentive, if somewhat creepy, custodian of the astronauts aboard the USS Discovery One, before famously turning homicidal and trying to kill them all. In The Matrix, humanity's invention of AI promptly results in human-machine warfare, leading to humans enslaved as a biological source of energy by the machines. In Daniel H. Wilson's book Robopocalypse, computer scientists finally crack the code on the AI problem, only to have their creation develop a sudden and deep dislike for its creators. Is Siri just a few upgrades away from killing you in your sleep? And you're not an especially sentient being yourself if you haven't heard the story of Skynet (see The Terminator, T2, T3, etc.) The simple answer is that -- movies like Wall-E, Short Circuit, and Chappie, notwithstanding -- Hollywood knows that nothing guarantees box office gold quite like an existential threat to all of humanity. Whether that threat is likely in real life or not is decidedly beside the point. How else can one explain the endless march of zombie flicks, not to mention those pesky, shark-infested tornadoes? The reality of AI is nothing like the movies. Siri, Alexa, Watson, Cortana -- these are our HAL 9000s, and none seems even vaguely murderous. The technology has taken leaps and bounds in the last decade, and seems poised to finally match the vision our artists have depicted in film for decades. Is Siri just a few upgrades away from killing you in your sleep, or is Hollywood running away with a tired idea? Looking back at the last decade of AI research helps to paint a clearer picture of a sometimes frightening, sometimes enlightened future. An increasing number of prominent voices are being raised about the real dangers of humanity's continuing work on so-called artificial intelligence.

A RealDoll model, not one currently equipped with AI. RealDoll, as the name suggests, make incredibly life-like sex dolls. You might have seen the documentary on the BBC about the four men who treat their RealDolls like girlfriends, or Lars and the Real Girl, the Ryan Gosling movie where the co-lead is a RealDoll named Bianca. Or you might have just seen them on the internet, for it is a vast and thorough thing. You also might have seen, on the internet, that RealDoll founder Matt McMullen is working on integrating robots and artificial intelligence into the dolls. While most AI personal assistants we chat to today--Siri, Microsoft's Cortana, Amazon's Echo--hire writers to ensure their responses are charming but always professional, Matt's trying to exploit the developments in speech recognition to create an AI whose main aim is to get you mentally and physically excited.

Google achieves that by pulling together the best features from Apple, Samsung and other phones and offering them at prices comparable to iPhones -- starting at about 650 for the regular, 5-inch model and 770 for the 5.5-inch "XL" edition. Both versions go on sale Thursday through Verizon, Best Buy and Google's online store. We tested the Pixel XL model; the regular version has identical features except for its smaller display and battery -- still enough for 13 hours of Internet use, according to Google. With either, you get an excellent camera and a strong voice assistant that promises to get smarter. The Pixel isn't quite an iPhone replacement, as Google wants you to believe; hardware is just part of what makes an iPhone an iPhone.

Google's fascination with hardware stretches back years. Remember the early days of Android and the G1? It took Google a while, but that fascination turned into a sort of experimental hobby, and now into something far more serious. Software is Google's art, and the company has been working for a long time to craft the right canvases. Google has more control over the development -- and destiny -- of these two smartphones than it ever had with any Nexus phone. It's not surprising, then, that the company has turned to close friends to help chart this new course. Former Motorola Mobility CEO Rick Osterloh is back at Google heading up hardware after the search giant sold his company to Lenovo. HTC, which most recently worked with Google on the Nexus 9 tablet, is handling the Pixel phones' production and assembly. There's a palpable sense that Google wanted to round up its A-team for this project.