We analyze properties of a map f sending a unitary matrix U of size N into a doubly stochastic matrix B = f (U) defined by Bi,j = |Ui,j| 2. For any U we define its defect, determined by the dimension of the image Df (TU U) of the space TU U tangent to the manifold of unitary matrices U at U under the tangent map Df corresponding to f. The defect, equal to zero for a generic unitary matrix, give...

Matrix eigen-decomposition is a classic and long-standing problem that plays a fundamental role in scientific computing and machine learning. Despite some existing algorithms for this inherently non-convex problem, the study remains inadequate for the need of large data nowadays. To address this gap, we propose a Doubly Stochastic Riemannian Gradient EIGenSolver, DSRG-EIGS, where the double sto...

As long as a square nonnegative matrix A contains sufficient nonzero elements, then the Sinkhorn-Knopp algorithm can be used to balance the matrix, that is, to find a diagonal scaling of A that is doubly stochastic. It is known that the convergence is linear and an upper bound has been given for the rate of convergence for positive matrices. In this paper we give an explicit expression for the ...

Let T be an arbitrary n × n matrix with real entries. We consider the set of all matrices with a given complex number as an eigenvalue, as well as being given the corresponding left and right eigenvectors. We find the closest matrix A, in Frobenius norm, in this set to the matrix T . The normal cone to a matrix in this set is also obtained. We then investigate the problem of determining the clo...

This paper presents a scalable distributed algorithm for computing and maintaining multi-target identity information. The algorithm builds on a novel representational framework, Identity-Mass Flow, to overcome the problem of exponential computational complexity in managing multi-target identity explicitly. The algorithm uses local information to efficiently update the global multi-target identi...

Preparation First, based on the definitions of A t , Y t , ˜ Z t and Z t , we can write g t = G(s t , r t , X t) = p −1 st p −1 rt (I − X t X ⊤ t)(E st ⊙ A)(E ·rt ⊙ X) = (I − X t X ⊤ t)A t Y t. Then from (6), we have X t+1 = X t + α t g t W t − α 2 t 2 X t g ⊤ t g t W t. Since W t = (I + α 2 t 4 g ⊤ t g t) −1 = I − α 2 t 4 g ⊤ t g t + O(α 4 t), we get X t+1 = X t + α t A t Y t − α t X t X ⊤ t A...

1 Data Sources • Amazon abbreviates the AmazonBinary dataset in Chen’s collection. • Iris is from the UCI machine learning repository . • Votes is from the UCI machine learning repository . • ORL is from the Database of Faces of AT&T. • PIE is from CMU/VASC Image Database. • YaleB is from the Extended Yale Face Database B. • COIL20 is from Columbia University Image Library. • Isolet is from the...

A distributed Nash equilibrium seeking algorithm is presented for networked games. We assume an incomplete information available to each player about the other players’ actions. The players communicate over a strongly connected digraph to send/receive the estimates of the other players’ actions to/from the other local players according to a gossip communication protocol. Due to asymmetric infor...

Standard credit risk models rely on a doubly stochastic structure. Conditional on the evolution of common factors, defaults are independent. Recently, tests by Das, Duffie, Kapadia and Saita (2007) have cast doubt on the empirical validity of this assumption. We modify their estimation approach in two ways. First, we model intra-month patterns in observed defaults. Second, we estimate default i...

Establishing correspondence between features in two images of the same scene taken from different viewing angles is a challenging problem in image processing and computer vision. However, its solution is an important step in many applications like wide baseline stereo, 3D model alignment, creation of panoramic views etc.In this paper, we propose a technique for registration of two images of a f...