Matrix Completion via Max-Norm Constrained Optimization

Matrix completion has been well studied under the uniform sampling model and the trace-norm regularized methods perform well both theoretically and numerically in such a setting. However, the uniform sampling model is unrealistic for a range of applications and the standard trace-norm relaxation can behave very poorly when the underlying sampling scheme is non-uniform. In this paper we propose and analyze a max-norm constrained empirical risk minimization method for noisy matrix completion under a general sampling model. The optimal rate of convergence is established under the Frobenius norm loss in the context of approximately low-rank matrix reconstruction. It is shown that the max-norm constrained method is minimax rate-optimal and yields a unified and robust approximate recovery guarantee, with respect to the sampling distributions. The computational effectiveness of this method is also discussed, based on first-order algorithms for solving convex optimizations involving max-norm regularization.Comment: 33 page

Paid Peering, Settlement-Free Peering, or Both?

With the rapid growth of congestion-sensitive and data-intensive applications, traditional settlement-free peering agreements with best-effort delivery often do not meet the QoS requirements of content providers (CPs). Meanwhile, Internet access providers (IAPs) feel that revenues from end-users are not sufficient to recoup the upgrade costs of network infrastructures. Consequently, some IAPs have begun to offer CPs a new type of peering agreement, called paid peering, under which they provide CPs with better data delivery quality for a fee. In this paper, we model a network platform where an IAP makes decisions on the peering types offered to CPs and the prices charged to CPs and end-users. We study the optimal peering schemes for the IAP, i.e., to offer CPs both the paid and settlement-free peering to choose from or only one of them, as the objective is profit or welfare maximization. Our results show that 1) the IAP should always offer the paid and settlement-free peering under the profit-optimal and welfare-optimal schemes, respectively, 2) whether to simultaneously offer the other peering type is largely driven by the type of data traffic, e.g., text or video, and 3) regulators might want to encourage the IAP to allocate more network capacity to the settlement-free peering for increasing user welfare

Nonlinear stability of the ensemble Kalman filter with adaptive covariance inflation

The Ensemble Kalman filter and Ensemble square root filters are data assimilation methods used to combine high dimensional nonlinear models with observed data. These methods have proved to be indispensable tools in science and engineering as they allow computationally cheap, low dimensional ensemble state approximation for extremely high dimensional turbulent forecast models. From a theoretical perspective, these methods are poorly understood, with the exception of a recently established but still incomplete nonlinear stability theory. Moreover, recent numerical and theoretical studies of catastrophic filter divergence have indicated that stability is a genuine mathematical concern and can not be taken for granted in implementation. In this article we propose a simple modification of ensemble based methods which resolves these stability issues entirely. The method involves a new type of adaptive covariance inflation, which comes with minimal additional cost. We develop a complete nonlinear stability theory for the adaptive method, yielding Lyapunov functions and geometric ergodicity under weak assumptions. We present numerical evidence which suggests the adaptive methods have improved accuracy over standard methods and completely eliminate catastrophic filter divergence. This enhanced stability allows for the use of extremely cheap, unstable forecast integrators, which would otherwise lead to widespread filter malfunction.Comment: 34 pages. 4 figure

Coexistence of Localized and Extended States in Disordered Systems

It is commonly believed that Anderson localized states and extended states do not coexist at the same energy. Here we propose a simple mechanism to achieve the coexistence of localized and extended states in a band in a class of disordered quasi-1D and quasi-2D systems. The systems are partially disordered in a way that a band of extended states always exists, not affected by the randomness, whereas the states in all other bands become localized. The extended states can overlap with the localized states both in energy and in space, achieving the aforementioned coexistence. We demonstrate such coexistence in disordered multi-chain and multi-layer systems.Comment: 5 pages, 3 figure