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Trace norm regularization is a widely used approach for learning low rank matrices. A standard optimization strategy is based on formulating the problem as one of low rank matrix factorization which, however, leads to a non-convex problem. In practice this approach works well, and it is often computationally faster than standard convex solvers such as proximal gradient methods. Nevertheless, it is not guaranteed to converge to a global optimum, and the optimization can be trapped at poor stationary points. In this paper we show that it is possible to characterize all critical points of the non-convex problem. This allows us to provide an efficient criterion to determine whether a critical point is also a global minimizer. Our analysis suggests an iterative meta-algorithm that dynamically expands the parameter space and allows the optimization to escape any non-global critical point, thereby converging to a global minimizer. The algorithm can be applied to problems such as matrix completion or multitask learning, and our analysis holds for any random initialization of the factor matrices. Finally, we confirm the good performance of the algorithm on synthetic and real datasets.

Summary: Decisions, including social decisions, are ultimately expressed through actions. However, very little is known about the kinematics of social decisions, and whether movements might reveal important aspects of social decision-making. We addressed this question by developing a motor version of a widely used behavioral economic game - the Ultimatum Game - and using a multivariate kinematic decoding approach to map parameters of social decisions to the single-trial kinematics of individual responders. Using this approach, we demonstrated that movement contains predictive information about both the fairness of a proposed offer and the choice to either accept or reject that offer. This information is expressed in personalized kinematic patterns that are consistent within a given responder, but that varies from one responder to another. These results provide insights into the relationship between decision-making and sensorimotor control, as they suggest that hand kinematics can reveal hidden parameters of complex, social interactive, choice.

Abstract Failure to develop prospective motor control has been proposed to be a core phenotypic marker of autism spectrum disorders (ASD). However, whether genuine differences in prospective motor control permit discriminating between ASD and non-ASD profiles over and above individual differences in motor output remains unclear. Here, we combined high precision measures of hand movement kinematics and rigorous machine learning analyses to determine the true power of prospective movement data to differentiate children with autism and typically developing children. Our results show that while movement is unique to each individual, variations in the kinematic patterning of sequential grasping movements genuinely differentiate children with autism from typically developing children. These findings provide quantitative evidence for a prospective motor control impairment in autism and indicate the potential to draw inferences about autism on the basis of movement kinematics.

Simulating the time-evolution of quantum mechanical systems is BQP-hard and expected to be one of the foremost applications of quantum computers. We consider classical algorithms for the approximation of Hamiltonian dynamics using subsampling methods from randomized numerical linear algebra. We derive a simulation technique whose runtime scales polynomially in the number of qubits and the Frobenius norm of the Hamiltonian. As an immediate application, we show that sample based quantum simulation, a type of evolution where the Hamiltonian is a density matrix, can be efficiently classically simulated under specific structural conditions. Our main technical contribution is a randomized algorithm for approximating Hermitian matrix exponentials. The proof leverages a low-rank, symmetric approximation via the Nyström method. Our results suggest that under strong sampling assumptions there exist classical poly-logarithmic time simulations of quantum computations.

The advent of geographic online social networks such as Foursquare, where users voluntarily signal their current location, opens the door to powerful studies on human movement. In particular the fine granularity of the location data, with GPS accuracy down to 10 meters, and the worldwide scale of Foursquare adoption are unprecedented. In this paper we study urban mobility patterns of people in several metropolitan cities around the globe by analyzing a large set of Foursquare users. Surprisingly, while there are variations in human movement in different cities, our analysis shows that those are predominantly due to different distributions of places across different urban environments. Moreover, a universal law for human mobility is identified, which isolates as a key component the rank-distance, factoring in the number of places between origin and destination, rather than pure physical distance, as considered in some previous works. Building on our findings, we also show how a rank-based movement model accurately captures real human movements in different cities.

Protein-protein interactions are critically dependent on just a few 'hot spot' residues at the interface. Hot spots make a dominant contribution to the free energy of binding and they can disrupt the interaction if mutated to alanine. Here, we present HSPred, a support vector machine(SVM)-based method to predict hot spot residues, given the structure of a complex. HSPred represents an improvement over a previously described approach (Lise et al, BMC Bioinformatics 2009, 10:365). It achieves higher accuracy by treating separately predictions involving either an arginine or a glutamic acid residue. These are the amino acid types on which the original model did not perform well. We have therefore developed two additional SVM classifiers, specifically optimised for these cases. HSPred reaches an overall precision and recall respectively of 61% and 69%, which roughly corresponds to a 10% improvement. An implementation of the described method is available as a web server at http://bioinf.cs.ucl.ac.uk/hspred. It is free to non-commercial users.