Can we recover a complex signal from its Fourier magnitudes? More generally, given a set of $m$ measurements, $y_k = |\mathbf a_k^* \mathbf x|$ for $k = 1, \dots, m$, is it possible to recover $\mathbf x \in \mathbb{C}^n$ (i.e., length-$n$ complex vector)? This **generalized phase retrieval** (GPR) problem is a fundamental task in various disciplines, and has been the subject of much recent investigation. Natural nonconvex heuristics often work remarkably well for GPR in practice, but lack clear theoretical explanations. In this paper, we take a step towards bridging this gap. We prove that when the measurement vectors $\mathbf a_k$'s are generic (i.i.d. complex Gaussian) and the number of measurements is large enough ($m \ge C n \log^3 n$), with high probability, a natural least-squares formulation for GPR has the following benign geometric structure: (1) there are no spurious local minimizers, and all global minimizers are equal to the target signal $\mathbf x$, up to a global phase; and (2) the objective function has a negative curvature around each saddle point. This structure allows a number of iterative optimization methods to efficiently find a global minimizer, without special initialization. To corroborate the claim, we describe and analyze a second-order trust-region algorithm.

Artificial intelligence (AI) has officially gone mainstream. Industry research firm Gartner named AI as its number one strategic technology for a second year in a row. The acquisitions race among giants like Google, IBM, Salesforce and Apple to purchase private AI companies keeps heating up -- 2016 alone saw 40 AI-related acquisitions and our own research found that 62% of large enterprises will be using AI-technologies by 2018. Since everyone seems to be talking about AI broadly, we at Narrative Science*–where we work with enterprises to close the communication gap between man and machine– focused our predictions this year on what we see happening with communications and AI. For 2017, we predict changes in to how we'll communicate with computers and other devices, how AI systems will communicate with each other, and how we'll communicate with each other about AI. The recent, combined efforts of a number of innovative tech giants point to a coming year when interacting with technology through conversation becomes the norm.

Foxconn, the Taiwanese manufacturing giant behind Apple's iPhone and numerous other major electronics devices, aims to automate away a vast majority of its human employees, according to a report from DigiTimes. Dai Jia-peng, the general manager of Foxconn's automation committee, says the company has a three-phase plan in place to automate its Chinese factories using software and in-house robotics units, known as Foxbots. The first phase of Foxconn's automation plans involve replacing the work that is either dangerous or involves repetitious labor humans are unwilling to do. The second phase involves improving efficiency by streamlining production lines to reduce the number of excess robots in use. The third and final phase involves automating entire factories, "with only a minimal number of workers assigned for production, logistics, testing, and inspection processes," according to Jia-peng.

In November, WIRED published a special issue guest-edited by President Obama. The magazine's features editor Maria Streshinsky says that working with the president was an exciting opportunity for everyone at WIRED, especially editor in chief Scott Dadich. "He could really recognize a lot of the language that the president would use as far as what the future could hold, and that it's well within our grasp to have an optimistic future," Streshinsky says in Episode 236 of the Geek's Guide to the Galaxy podcast. "Those are the ideas that the president was very interested in, and that just sit squarely in what Scott believes and what WIRED tries to focus on." WIRED associate editor Jason Kehe, a big science fiction fan, was particularly excited to learn more about the president's taste in science fiction.

I will update this article regularly. An old version can be found here and has many interesting links. All the material presented here is not in the old version. This article is divided into 11 sections. A laptop is the ideal device.

Then consider this month's winners of the 2016 Siemens Competition, which honors math, science and technology projects from high school students around the country. These are some smart kids with plans to revolutionize fields such as medicine. Identical twin sisters from Texas won the $100,000 prize in the team event with their project that delivers an earlier diagnosis for schizophrenia. The $100,000 scholarship winner of the individual competition is from Oregon and he developed a biodegradable battery to power medical devices that you swallow. The sisters, Adhya and Shriya Beesam, are juniors (yes, juniors) at Plano East Senior High School in Plano, Texas, north of Dallas.

Salesforce ( CRM) continued its stellar performance in 2016, with its top line growing at more than 25% in the first three quarters of the fiscal year and beating market expectations. Additionally, the company's added focus on improving its bottom line started to pay dividends, with its earnings per share for the first nine months growing appreciably from -$0.03 in 2015 to $0.34 in 2016. Moreover, during the year, Salesforce made a number of acquisitions and made a push in the e-commerce and artificial intelligence domains, opening up avenues for further growth. Apart from this, the company also affirmed its goal of $10 billion in revenues, which it expects to achieve by the end of next year. Despite a good performance in the first three quarters of the year, Salesforce's stock is currently trading 12% lower than its price in January, owing to a tougher market environment and relatively soft performance in the second quarter.

Amazon's 2016 has been record breaking on many fronts. The company recorded its sixth consecutive quarterly profit (previously, it mostly hemorrhaged cash). Meanwhile, this year marked Amazon's growing strength in hardware with its hit Echo home automation hub Amazon Echo, and its companion voice assistant Alexa. The company has also become force in entertainment, debuting a line of hit original shows through its Amazon Video Prime service. It's hard to imagine how Amazon could top 2016, but here are some likely moves by the Seattle-based Goliath in 2017: To save money over the past year, Amazon has been seeking to take over more shipping duties from the likes of UPS and FedEx by leasing trucks, planes, and ships.

Artificial intelligence (AI) has officially gone mainstream. Industry research firm Gartner named AI as its number one strategic technology for a second year in a row. The acquisitions race among giants like Google, IBM, Salesforce and Apple to purchase private AI companies keeps heating up -- 2016 alone saw 40 AI-related acquisitions and our own research found that 62% of large enterprises will be using AI-technologies by 2018. Since everyone seems to be talking about AI broadly, we focused our predictions this year on what we see happening with communications and AI. See also: Will artificial intelligence mean the end of cyberthreats?

This is a follow up to an article Per wrote last year, Machine Learning in a Week, on how he kickstarted his way into machine learning (ml) by devoting five days to the subject. Follow him on Medium and check out his archive. My interest in ml stems back to 2014 when I started reading articles about it on Hacker News. I simply found the idea of teaching machines stuff by looking at data appealing. At the time I wasn't even a professional developer, but a hobby coder who'd done a couple of small projects. So I began watching the first few chapters of Udacity's Supervised Learning course, while also reading all articles I came across on the subject.