We derive a Newton method for computing the best rank-(r 1 , r 2 , r 3) approximation of a given J × K × L tensor A. The problem is formulated as an approximation problem on a product of Grassmann manifolds. Incorporating the manifold structure into Newton's method ensures that all iterates generated by the algorithm are points on the Grassmann manifolds. We also introduce a consistent notation for matricizing a tensor, for contracted tensor products and some tensor-algebraic manipulations, which simplify the derivation of the Newton equations and enable straightforward algorithmic implementation. Experiments show a quadratic convergence rate for the Newton-Grassmann algorithm.

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In this paper we proposed quasi-Newton and limited memory quasi-Newton methods for objective functions defined on Grassmannians or a product of Grassmannians. Specifically we defined bfgs and l-bfgs updates in local and global coordinates on Grassmannians or a product of these. We proved that, when local coordinates are used, our bfgs updates on Grassmannians share the same optimality property as the usual bfgs updates on Euclidean spaces. When applied to the best multilinear rank approximation problem for general and symmetric tensors, our approach yields fast, robust, and accurate algorithms that exploit the special Grassmannian structure of the respective problems, and which work on tensors of large dimensions and arbitrarily high order.Extensive numerical experiments are included to substantiate our claims.The second problem is that of finding a best multilinear rank-r approximation to a symmetric tensor S ∈ S 3 (R n ). In other words, we seek a matrix Q whose columns are mutually orthonormal, and a symmetric tensor C ∈ S 3 (R r ) such that a multilinear transformation of C by Q approximates S in the sense of minimizing a sum-of-squares loss. Using the same notation as in (1.1), the problem is argmin Q∈O(n,r),C∈S 3 (R r ) S − (Q, Q, Q) · C F .( 1.2)This problem is significant because many important tensors that arise in applications are symmetric tensors.

Abstract-Link prediction is a fundamental problem in social network analysis and modern-day commercial applications such as Facebook and Myspace. Most existing research approaches this problem by exploring the topological structure of a social network using only one source of information. However, in many application domains, in addition to the social network of interest, there are a number of auxiliary social networks and/or derived proximity networks available. The contribution of the paper is twofold: (1) a supervised learning framework that can effectively and efficiently learn the dynamics of social networks in the presence of auxiliary networks; (2) a feature design scheme for constructing a rich variety of path-based features using multiple sources, and an effective feature selection strategy based on structured sparsity. Extensive experiments on three real-world collaboration networks show that our model can effectively learn to predict new links using multiple sources, yielding higher prediction accuracy than unsupervised and singlesource supervised models.

In this paper we present a fast and accurate procedure called clustered low rank matrix approximation for massive graphs. The procedure involves a fast clustering of the graph and then approximates each cluster separately using existing methods, e.g. the singular value decomposition, or stochastic algorithms. The cluster-wise approximations are then extended to approximate the entire graph. This approach has several benefits: (1) important community structure of the graph is preserved due to the clustering; (2) highly accurate low rank approximations are achieved; (3) the procedure is efficient both in terms of computational speed and memory usage; (4) better performance in problems from various applications compared to standard low rank approximation. Further, we generalize stochastic algorithms to the clustered low rank approximation framework and present theoretical bounds for the approximation error. Finally, a set of experiments, using large scale and real-world graphs, show that our methods outperform standard low rank matrix approximation algorithms.