Microsoft is reportedly developing a version of its year-old digital assistant, named Cortana, for iOS and Android. The move seems to indicate Microsoft is trying new ways to get its products onto the two dominant mobile operating systems across the US. For mobile users, the question will be: how does Cortana stack up next to Siri? The Cortana project came out of a branch of Microsoft developing artificial intelligence, dubbed "Einstein." Cortana, which is named after a central character in the massively popular Microsoft-owned video game series Halo, has been installed on Microsoft phones for the past year.

Apple personal virtual assistant, Siri, might soon be competing for your time with Cortana, its counterpart from Microsoft. Cortana will be coming to iOS and Android at some point after Windows 10 rolls out with an updated version of Microsoft's virtual assistant software, Reuters reported Friday, citing people who claim to have knowledge of the software giant's plans. It would be a standalone app, available in the Google Play marketplace and Apple App Store, and work just as it already does on Windows Phone, according to the report. Microsoft also is working toward a more advanced version of Cortana, drawing from a research project called Einstein. "This kind of technology, which can read and understand email, will play a central role in the next rollout of Cortana, which we are working on now for the fall time frame," Eric Horvitz, Microsoft Research managing director, told Reuters in an interview. The company has already incorporated Cortana into its Windows 10 operating system, which will be coming to PCs in the latter part of this year.

Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study the statistical limits of convex relaxations. Particularly, we consider two problems: Mean estimation for sparse principal submatrix and edge probability estimation for stochastic block model. We exploit the sum-of-squares relaxation hierarchy to sharply characterize the limits of a broad class of convex relaxations. Our result shows statistical optimality needs to be compromised for achieving computational tractability using convex relaxations. Compared with existing results on computational lower bounds for statistical problems, which consider general polynomial-time algorithms and rely on computational hardness hypotheses on problems like planted clique detection, our theory focuses on a broad class of convex relaxations and does not rely on unproven hypotheses.

We consider two closely related problems: planted clustering and submatrix localization. The planted clustering problem assumes that a random graph is generated based on some underlying clusters of the nodes; the task is to recover these clusters given the graph. The submatrix localization problem concerns locating hidden submatrices with elevated means inside a large real-valued random matrix. Of particular interest is the setting where the number of clusters/submatrices is allowed to grow unbounded with the problem size. These formulations cover several classical models such as planted clique, planted densest subgraph, planted partition, planted coloring, and stochastic block model, which are widely used for studying community detection and clustering/bi-clustering. For both problems, we show that the space of the model parameters (cluster/submatrix size, cluster density, and submatrix mean) can be partitioned into four disjoint regions corresponding to decreasing statistical and computational complexities: (1) the \emph{impossible} regime, where all algorithms fail; (2) the \emph{hard} regime, where the computationally expensive Maximum Likelihood Estimator (MLE) succeeds; (3) the \emph{easy} regime, where the polynomial-time convexified MLE succeeds; (4) the \emph{simple} regime, where a simple counting/thresholding procedure succeeds. Moreover, we show that each of these algorithms provably fails in the previous harder regimes. Our theorems establish the minimax recovery limit, which are tight up to constants and hold with a growing number of clusters/submatrices, and provide a stronger performance guarantee than previously known for polynomial-time algorithms. Our study demonstrates the tradeoffs between statistical and computational considerations, and suggests that the minimax recovery limit may not be achievable by polynomial-time algorithms.

We obtain the conditions for the emergence of the swarm intelligence effect in an interactive game of restless multi-armed bandit (rMAB). A player competes with multiple agents. Each bandit has a payoff that changes with a probability $p_{c}$ per round. The agents and player choose one of three options: (1) Exploit (a good bandit), (2) Innovate (asocial learning for a good bandit among $n_{I}$ randomly chosen bandits), and (3) Observe (social learning for a good bandit). Each agent has two parameters $(c,p_{obs})$ to specify the decision: (i) $c$, the threshold value for Exploit, and (ii) $p_{obs}$, the probability for Observe in learning. The parameters $(c,p_{obs})$ are uniformly distributed. We determine the optimal strategies for the player using complete knowledge about the rMAB. We show whether or not social or asocial learning is more optimal in the $(p_{c},n_{I})$ space and define the swarm intelligence effect. We conduct a laboratory experiment (67 subjects) and observe the swarm intelligence effect only if $(p_{c},n_{I})$ are chosen so that social learning is far more optimal than asocial learning.

Investors in Sirius XM Holdings Inc (NASD: SIRI) saw new options become available today, for the May 1st expiration. At Stock Options Channel, our YieldBoost formula has looked up and down the SIRI options chain for the new May 1st contracts and identified the following call contract of particular interest. The call contract at the $4.00 strike price has a current bid of 6 cents. If an investor was to purchase shares of SIRI stock at the current price level of $3.90/share, and then sell-to-open that call contract as a "covered call," they are committing to sell the stock at $4.00. Considering the call seller will also collect the premium, that would drive a total return (excluding dividends, if any) of 4.10% if the stock gets called away at the May 1st expiration (before broker commissions).

This paper shows that one cannot learn the probability of rare events without imposing further structural assumptions. The event of interest is that of obtaining an outcome outside the coverage of an i.i.d. sample from a discrete distribution. The probability of this event is referred to as the "missing mass". The impossibility result can then be stated as: the missing mass is not distribution-free PAC-learnable in relative error. The proof is semi-constructive and relies on a coupling argument using a dithered geometric distribution. This result formalizes the folklore that in order to predict rare events, one necessarily needs distributions with "heavy tails".

We consider an enlarged dimension reduction space in functional inverse regression. Our operator and functional analysis based approach facilitates a compact and rigorous formulation of the functional inverse regression problem. It also enables us to expand the possible space where the dimension reduction functions belong. Our formulation provides a unified framework so that the classical notions, such as covariance standardization, Mahalanobis distance, SIR and linear discriminant analysis, can be naturally and smoothly carried out in our enlarged space. This enlarged dimension reduction space also links to the linear discriminant space of Gaussian measures on a separable Hilbert space.

Community detection is a fundamental problem in network analysis which is made more challenging by overlaps between communities which often occur in practice. Here we propose a general, flexible, and interpretable generative model for overlapping communities, which can be thought of as a generalization of the degree-corrected stochastic block model. We develop an efficient spectral algorithm for estimating the community memberships, which deals with the overlaps by employing the K-medians algorithm rather than the usual K-means for clustering in the spectral domain. We show that the algorithm is asymptotically consistent when networks are not too sparse and the overlaps between communities not too large. Numerical experiments on both simulated networks and many real social networks demonstrate that our method performs very well compared to a number of benchmark methods for overlapping community detection.

We describe various sets of conditional independence relationships, sufficient for qualitatively comparing non-vanishing squared partial correlations of a Gaussian random vector. These sufficient conditions are satisfied by several graphical Markov models. Rules for comparing degree of association among the vertices of such Gaussian graphical models are also developed. We apply these rules to compare conditional dependencies on Gaussian trees. In particular for trees, we show that such dependence can be completely characterized by the length of the paths joining the dependent vertices to each other and to the vertices conditioned on. We also apply our results to postulate rules for model selection for polytree models. Our rules apply to mutual information of Gaussian random vectors as well.