We give an algorithm for learning a permutation on-line. The algorithm maintains its uncertainty about the target permutation as a doubly stochastic matrix. This matrix is updated by multiplying the current matrix entries by exponential factors. These factors destroy the doubly stochastic property of the matrix and an iterative procedure is needed to re-normalize the rows and columns. Even thou...

In this paper we focus on the issue of normalization of the affinity matrix in spectral clustering. We show that the difference between N-cuts and Ratio-cuts is in the error measure being used (relative-entropy versus L1 norm) in finding the closest doubly-stochastic matrix to the input affinity matrix. We then develop a scheme for finding the optimal, under Frobenius norm, doubly-stochastic ap...

In this paper we propose a concavely regularized convex relaxation based graph matching algorithm. The graph matching problem is firstly formulated as a constrained convex quadratic program by relaxing the feasible set from the permutation matrices to doubly stochastic matrices. To gradually push the doubly stochastic matrix back to be a permutation one, an objective function is constructed by ...

A real or complex n × n matrix is generalized doubly stochastic if all of its row sums and column sums equal one. Denote by V the linear space spanned by such matrices. We study the reducibility of V under the group Γ of linear operators of the form A 7→ PAQ, where P and Q are n×n permutation matrices. Using this result, we show that every linear operator φ : V → V mapping the set of generalize...

This paper considers certain comparison techniques involving Markov chains with transition matrices Pa = aI+(1?a)P where P is the transition matrix of a doubly stochastic Markov chain. This paper provides upper bounds for how far the Markov chain with transition matrix Pb is from uniformly distributed after n steps. These upper bounds involve how far the Markov chain with transition matrix Pa i...

A sharp lower bound for the smallest entries, among those corresponding to edges, of doubly stochastic matrices of trees is obtained, and the trees that attain this bound are characterized. This result is used to provide a negative answer to Merris’ question in [R. Merris, Doubly stochastic graph matrices II, Linear Multilin. Algebra 45 (1998) 275–285]. © 2005 Elsevier Inc. All rights reserved....

The permanent function is used to determine geometrical properties of the set 52, of all II x it nonnegative doubly stochastic matrices. If ,F is a face of Q, , then F corresponds to an n x n (0, I)-matrix A, where the permanent of A is the number of vertices of 3. I f A is fully indecomposable, then the dimension of 9 equals u(A) 2n + 1, where u(A) is the number of I’s in A. The only twodimens...

Introduction. A classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and nea...

A multiple linear process with random coefficients is investigated in the paper. Conditions for existence of such process are derived and its covariance function as well as the matrix of spectral densities are calculated. The results are applied to multiple AR(1) process with random coefficients, where the matrices of coefficients can be described by a stationary process. In this case condition...

Let X be a matrix sampled uniformly from the set of doubly stochastic matrices of size n×n. We show that the empirical spectral distribution of the normalized matrix √ n(X − EX) converges almost surely to the circular law. This confirms a conjecture of Chatterjee, Diaconis and Sly.