Cluster analysis by nonnegative low-rank approximations has experienced a remarkable progress in the past decade. However, the majority of such approximation approaches are still restricted to nonnegative matrix factorization (NMF) and suffer from the following two drawbacks: 1) they are unable to produce balanced partitions for large-scale manifold data which are common in real-world clusterin...

Moderated greedy search is based on the idea that it is helpful for greedy search algorithms to make non-optimal choices “once in a while.” This notion can be made precise by using the majorizationtheoretic approach to greedy algorithms. Majorization is the study of pre-orderings induced by doubly stochastic matrices. A majorization operator when applied to a distribution makes it “less unequal...

A stronger version of the van der Waerden permanent conjecture asserts that if J„ denotes the n X n matrix all of whose entries are \/n and A is any fixed matrix on the boundary of the set ofnxn doubly stochastic matrices, then per(A/4 + (1 — \)Jn) as a function of X is nondecreasing in the interval [0, 1]. In this paper, we elucidate the relation of this assertion to some other conjectures kno...

We study online combinatorial decision problems, where one must make sequential decisions in some combinatorial space without knowing in advance the cost of decisions on each trial; the goal is to minimize the total regret over some sequence of trials relative to the best fixed decision in hindsight. Such problems have been studied mostly in settings where decisions are represented by Boolean v...

Many modern clustering methods employ a non-convex objective function and use iterative optimization algorithms to find local minima. Thus initialization of the algorithms is very important. Conventionally the starting guess of the iterations is randomly chosen; however, such a simple initialization often leads to poor clusterings. Here we propose a new method to improve cluster analysis by com...

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...