Doubly-stochastic Matrices Research Articles

On the trace-zero doubly stochastic matrices of order 5

We propose a graph theoretic approach to determine the trace of the product of two permutation matrices through a weighted digraph representation for a pair of permutation matrices. Consequently, we derive trace-zero doubly stochastic (DS) matrices of order 5 whose k-th power is also a trace-zero DS matrix for k∈{2,3,4,5}. Then, we determine necessary conditions for the coefficients of a generic polynomial of degree 5 to be realizable as the characteristic polynomial of a trace-zero DS matrix of order 5. Finally, we approximate the eigenvalue region of trace-zero DS matrices of order 5.

Visual Permutation Learning.

We present a principled approach to uncover the structure of visual data by solving a deep learning task coined visual permutation learning. The goal of this task is to find the permutation that recovers the structure of data from shuffled versions of it. In the case of natural images, this task boils down to recovering the original image from patches shuffled by an unknown permutation matrix. Permutation matrices are discrete, thereby posing difficulties for gradient-based optimization methods. To this end, we resort to a continuous approximation using doubly-stochastic matrices and formulate a novel bi-level optimization problem on such matrices that learns to recover the permutation. Unfortunately, such a scheme leads to expensive gradient computations. We circumvent this issue by further proposing a computationally cheap scheme for generating doubly stochastic matrices based on Sinkhorn iterations. To implement our approach we propose DeepPermNet, an end-to-end CNN model for this task. The utility of DeepPermNet is demonstrated on three challenging computer vision problems, namely, relative attributes learning, supervised learning-to-rank, and self-supervised representation learning. Our results show state-of-the-art performance on the Public Figures and OSR benchmarks for relative attributes learning, chronological and interestingness image ranking for supervised learning-to-rank, and competitive results in the classification and segmentation tasks of the PASCAL VOC dataset for self-supervised representation learning.

Ehrhart Series of Polytopes Related to Symmetric Doubly-Stochastic Matrices

In Ehrhart theory, the $h^*$-vector of a rational polytope often provides insights into properties of the polytope that may be otherwise obscured. As an example, the Birkhoff polytope, also known as the polytope of real doubly-stochastic matrices, has a unimodal $h^*$-vector, but when even small modifications are made to the polytope, the same property can be very difficult to prove. In this paper, we examine the $h^*$-vectors of a class of polytopes containing real doubly-stochastic symmetric matrices.

On spectral properties for graph matching and graph isomorphism problems

Problems related to graph matching and isomorphisms are very important both from a theoretical and practical perspective, with applications ranging from image and video analysis to biological and biomedical problems. The graph matching problem is challenging from a computational point of view, and therefore different relaxations are commonly used. Although common relaxations techniques tend to work well for matching perfectly isomorphic graphs, it is not yet fully understood under which conditions the relaxed problem is guaranteed to obtain the correct answer. In this paper, we prove that the graph matching problem and its most common convex relaxation, where the matching domain of permutation matrices is substituted with its convex hull of doubly-stochastic matrices, are equivalent for a certain class of graphs, such equivalence being based on spectral properties of the corresponding adjacency matrices. We also derive results about the automorphism group of a graph, and provide fundamental spectral properties of the adjacency matrix.

Tropical determinant of integer doubly-stochastic matrices

Let D(m,n) be the set of all the integer points in the m-dilate of the Birkhoff polytope of doubly-stochastic n×n matrices. In this paper we find the sharp upper bound on the tropical determinant over the set D(m,n). We define a version of the tropical determinant where the maximum over all the transversals in a matrix is replaced with the minimum and then find the sharp lower bound on thus defined tropical determinant over D(m,n).

Convex solution of a permutation problem

In this paper, we show that a problem of finding a permuted version of k vectors from RN such that they belong to a prescribed rank r subset, can be solved by convex optimization. We prove that under certain generic conditions, the wanted permutation matrix is unique in the convex set of doubly-stochastic matrices. In particular, this implies a solution of the classical correspondence problem of finding a permutation that transforms one collection of points in Rk into the another one. Solutions to these problems have a wide set of applications in Engineering and Computer Science.

A generating function for all semi-magic squares and the volume of the Birkhoff polytope

We present a multivariate generating function for all n×n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope B n of n×n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of B n and any of its faces.

A characterization of even doubly-stochastic matrices

Even doubly-stochastic matrices are characterized with the aid of the minima of functionals defined by the even diagonals contained in the matrix.

An inverse problem for symmetric doubly stochastic matrices

In this paper, we study the inverse eigenvalue problem for n × n symmetricdoubly stochastic matrices. The spectra of all indecomposable imprimitive symmetric doublystochastic matrices are characterized. Then we obtain new sufficient conditions for a realn-tuple to be thespectrum of an n × nsymmetric doubly stochastic matrix of zero trace. Also, we provethat the set where the decreasingly ordered spectra of all n × nsymmetric doubly stochastic matrices lie is not convex. As a consequence, weprove that the set where the decreasingly ordered spectra of all n × nnon-negative matrices lie is not convex.

Birkhoff's Theorem for Panstochastic Matrices

The panstochastic analogue of Birkhoff's Theorem on doubly-stochastic matrices is proved in the case $n=5$. It is shown that this analogue fails when $n>1$, $n \ne 5$.

On some permanental conjectures

Let Ω n denote the set of all n×n doubly-stochastic matrices and let σ k (A) be the sum of all subpermanents of order k of matrix A. We prove the Holens-Dokovic conjecture for k=4 and n≥5. Namely, Let be th set of all matrices from Ω n with zero main diagonal and let be the matrix from with 1/(n−1) in its off-diagonal positions. We prove the following for 2≤k≤4 and n≥k+1: for any A consequence of the last inequality and also of a result of D. London and H. Minc [8] is the inequality .

Entropy, information and doubly stochastic transformations

I analyze two inequalities on entropy and information, one due to von Neumann and a recent one to Schiffer, and show that the relevant quantities in these inequalities are related by special doubly stochastic matrices (DSM). I then use generalization of the first inequality to prove algebraically a generalization of Schiffer's inequality to arbitrary DSM. I also give a second interpretation to the latter inequality, determine its domain of applicability, and illustrate it by using Zeeman splitting. This example shows that symmetric (degenerate) systems have less entropy than the corresponding split systems, if compared at the same average energy. This seemingly counter-intuitive result is explained thermodynamically.

Expanders and Diffusers

Expander graphs are ingredients for making concentrating, switching, and sorting networks, and are closely related to sparse, doubly-stochastic matrices called diffusers. The best explicit examples of diffusers are defined by means of the action of elements of the matrix group $SL (2,{\bf Z} )$ on certain finite mathematical objects. Some corresponding, explicit expanders were introduced by Margulis. However, Gabber and Galil were the first to obtain good estimates for the expanders and produce from them a family of directed acyclic superconcentrators having density 271.8. In this paper we review various techniques for making expanders from diffusers. We also demonstrate asymptotic upper bounds on the strength of algebraically defined classes of degree k diffusers. Each upper bound is given as the norm of a diffusion operator on an infinite discrete group, and bounds for several examples are calculated. Numerical evidence is supplied in support of our conjecture that these bounds can be achieved by certain algebraically defined examples. The conjecture, if true, would lead to superconcentrators of density less than 58.

On a theorem of L. Mirsky on even doubly-stochastic matrices

A necessary condition for a matrix to be an even doubly-stochastic one, found by L. Mirsky in 1961, is shown to be not sufficient.

The harmonic means of diagonals of doubly-stochastic matrices

It is shown that every doubly-stochastic n × n matrix with precisely r positive elements possesses a positive diagonal whose harmonic mean is at least equal to n/r.

A note on 1-factors in certain regular multigraphs

Claude Berge has given sufficient conditions under which each edge of a regular multigraph belongs to some 1-factor. His proof-technique actually yields the same conclusion under less restrictive conditions. Here we point out that this stronger version of Berge's theorem follows easily from a theorem on doubly-stochastic matrices proved previously by the author.

A note on symmetric doubly-stochastic matrices

Each extreme point in the convex set Δ∗n of all n×n symmetric doubly-stochastic matrices is shown to have the form 12(P + Pt), for some n×n permutation matrix P. The convex hull Σn of the integral points in Δ∗n (i.e., the symmetric permutation matrices) is shown to consist of those matrices, X = (xij) in Δ∗n satisfying Σi∈SΣj∈S−{i}xij ⩽ 2k, for each subset S of {1, 2, …, n} having cardinality 2k + 1, for some k > 0.

Correction to: “A Note on Doubly-Stochastic Matrices”

Correction to: “A Note on Doubly-Stochastic Matrices”

The semigroup of doubly-stochastic matrices

The set Dn of all n × n doubly-stochastic matrices is a semigroup with respect to ordinary matrix multiplication. This note is concerned with the determination of the maximal subgroups of Dn. It is shown that the number of subgroups is finite, that each subgroup is finite and is in fact isomorphic to a direct product of symmetric groups. These results are applied in § 3 to yield information about the least number of permutation matrices whose convex hull contains a given doubly-stochastic matrix.

Spectral properties of doubly-stochastic matrices

Spectral properties of doubly-stochastic matrices