International Workshop on Numerical Linear Algebra with Applications

A matrix ! is completely positive if it has a nonnegative factorization ! = !!! where ! is element wise nonnegative. The smallest number of columns in such a matrix ! is called the cp-rank of !. The !×! completely positive matrices form a closed convex cone and its dual is the cone of co-positive matrices. A completely positive matrix is doubly nonnegative – it is positive semi definite and it is element wise nonnegative. The !×! doubly nonnegative matrices also form a closed convex cone which, for ! > 4, strictly contains the cone of completely positive matrices. Every optimization problem with quadratic objective function, linear constrains and binary variables can be equivalently written as a linear problem over the completely positive cone. However, the problem of checking whether a given matrix is completely positive is NP-hard. A bound for the optimal value can be obtained by replacing the completely positive cone by the doubly nonnegative cone. In the talk I will show how one can generate co-positive cuts for this relaxation. I will also describe some recent results on the cp-rank. Using Large-scale Matrix Factorizations to Identify Users of Social Networks Michael W. Berry Department of Electrical Engineering and Computer Science The University of Tennessee Abstract Users of social media interact with the network and its users. Each interaction creates network specific data between the engaged users and the chosen social avenue. Over time, these engagements accumulate to describe the user's social fingerprint, an identity which encapsulates the user's persona on the network. The agglomeration of this information showcases the user's activity on the social network and establishes a traceable social fingerprint. These fingerprints can be tracked and stored as large matrices representing the quantity and occurrence of observed user behavior. We seek to apply large-scale matrix factorization techniques to establish the signature component vector of a social network user's identity. The preliminary results presented will demonstrate that a user's social finger-print is both quantifiable and identifiable on a social network through out time.Users of social media interact with the network and its users. Each interaction creates network specific data between the engaged users and the chosen social avenue. Over time, these engagements accumulate to describe the user's social fingerprint, an identity which encapsulates the user's persona on the network. The agglomeration of this information showcases the user's activity on the social network and establishes a traceable social fingerprint. These fingerprints can be tracked and stored as large matrices representing the quantity and occurrence of observed user behavior. We seek to apply large-scale matrix factorization techniques to establish the signature component vector of a social network user's identity. The preliminary results presented will demonstrate that a user's social finger-print is both quantifiable and identifiable on a social network through out time. This is a joint work with Denise R. Koessler.