Related papers: Matrix scaling and explicit doubly stochastic limi…
The process of alternately row scaling and column scaling a positive $n \times n$ matrix $A$ converges to a doubly stochastic positive $n \times n$ matrix $S(A)$, called the \emph{Sinkhorn limit} of $A$. Exact formulae for the Sinkhorn…
The alternate row and column scaling algorithm applied to a positive $n\times n$ matrix $A$ converges to a doubly stochastic matrix $S(A)$, sometimes called the \emph{Sinkhorn limit} of $A$. For every positive integer $n$, a two parameter…
Applied to a nonnegative $m\times n$ matrix with a nonzero $\sigma$-diagonal, the sequence of matrices constructed by alternate row and column scaling conveges to a doubly stochastic matrix. It is proved that if this sequence converges…
Sinkhorn's alternative minimization algorithm applied to a positive $n\times n$ matrix converges to a doubly stochastic matrix. If the algorithm, applied to a $2\times 2$ matrix, converges in a finite number of iterations, then it converges…
The Kruithof iterative scaling process, which adjusts matrices to meet target row and column sums, is a longstanding problem that lacks a general closed form for its limit. While Nathanson derived the closed form for the Sinkhorn limit of…
Given a nonnegative matrix $A$, can you find diagonal matrices $D_1,~D_2$ such that $D_1AD_2$ is doubly stochastic? The answer to this question is known as Sinkhorn's theorem. It has been proved with a wide variety of methods, each…
The iterative method of Sinkhorn allows, starting from an arbitrary real matrix with non-negative entries, to find a so-called 'scaled matrix' which is doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and with all…
We develop several efficient algorithms for the classical \emph{Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input $n\times n$ matrix $A$, this…
For a given nonnegative matrix $A=(A_{ij})$, the matrix scaling problem asks whether $A$ can be scaled to a doubly stochastic matrix $D_1AD_2$ for some positive diagonal matrices $D_1,D_2$.The Sinkhorn algorithm is a simple iterative…
The matrix scaling problem, particularly the Sinkhorn-Knopp algorithm, has been studied for over 60 years. In practice, the algorithm often yields high-quality approximations within just a few iterations. Theoretically, however, the…
It is well known that any positive matrix can be scaled to have prescribed row and column sums by multiplying its rows and columns by certain positive scaling factors (which are unique up to a positive scalar). This procedure is known as…
Sinkhorn proved that every entry-wise positive matrix can be made doubly stochastic by multiplying with two diagonal matrices. In this note we prove a recently conjectured analogue for unitary matrices: every unitary can be decomposed into…
In this paper we show how to construct diagonal scalings for arbitrary matrix pencils $\lambda B-A$, in which both $A$ and $B$ are complex matrices (square or nonsquare). The goal of such diagonal scalings is to "balance" in some sense the…
Tensor models generalize matrix models and generate colored triangulations of pseudo-manifolds in dimensions $D\geq 3$. The free energies of some models have been recently shown to admit a double scaling limit, i.e. large tensor size $N$…
In 2013, Cuturi [Cut13] introduced the Sinkhorn algorithm for matrix scaling as a method to compute solutions to regularized optimal transport problems. In this paper, aiming at a better convergence rate for a high accuracy solution, we…
We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate the limiting correlation functions to a nonlinear hierarchy of ordinary differential equations.
Covariances and variances of linear statistics of a point process can be written as integrals over the truncated two-point correlation function. When the point process consists of the eigenvalues of a random matrix ensemble, there are often…
We use a variety of computational tools to obtain a degree-$\binom{m + n - 2}{m - 1}$ polynomial equation conjecturally satisfied by the top-left entry of the Sinkhorn limit of a positive $m \times n$ matrix. The degree of this equation has…
Birkhoff polytope is the set of all bistochastic matrices (also known as doubly stochastic matrices). Bistochastic matrices form a special class of stochastic matrices where each row and column sums up to one. Permutation matrices and…
Matrix scaling and matrix balancing are two basic linear-algebraic problems with a wide variety of applications, such as approximating the permanent, and pre-conditioning linear systems to make them more numerically stable. We study the…