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Semi-supervised learning based on manifolds has been the focus of extensive research in recent years. Convenient neighbourhood graph construction is a key component of a successful semi-supervised classification method. Previous graph construction methods fail when there are pairs of data points that have small Euclidean distance, but are far apart over the manifold. To overcome this problem, we start with an arbitrary neighbourhood graph and iteratively update the edge weights by using the estimates of the geodesic distances between points. Moreover, we provide theoretical bounds on the values of estimated geodesic distances. Experimental results on real-world data show significant... 

This letter considers the use of total variation (TV) minimization in the recovery of a given gradient sparse vector from Gaussian linear measurements. It has been shown in recent studies that there exists a sharp phase transition behavior in TV minimization for the number of measurements necessary to recover the signal in asymptotic regimes. The phase-transition curve specifies the boundary of success and failure of TV minimization for large number of measurements. It is a challenging task to obtain a theoretical bound that reflects this curve. In this letter, we present a novel upper bound that suitably approximates this curve and is asymptotically sharp. Numerical results show that our... 

It is of fundamental importance to find algorithms obtaining optimal performance for learning of statistical models in distributed and communication limited systems. Aiming at characterizing the optimal strategies, we consider learning of Gaussian Processes (GP) in distributed systems as a pivotal example. We first address a very basic problem: how many bits are required to estimate the inner-products of some Gaussian vectors across distributed machines? Using information theoretic bounds, we obtain an optimal solution for the problem which is based on vector quantization. Two suboptimal and more practical schemes are also presented as substitutes for the vector quantization scheme. In... 

Semi-Supervised Learning (SSL) has become a topic of recent research that effectively addresses the problem of limited labeled data. Many SSL methods have been developed based on the manifold assumption, among them, the Local and Global Consistency (LGC) is a popular method. The problem with most of these algorithms, and in particular with LGC, is the fact that their naive implementations do not scale well to the size of data. Time and memory limitations are the major problems faced in large-scale problems. In this paper, we provide theoretical bounds on gradient descent, and to overcome the aforementioned problems, a new approximate Newton's method is proposed. Moreover, convergence...