A review of matrix scaling and Sinkhorn's normal form for matrices and positive maps (1609.06349v1)

Abstract: 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 presenting a variety of possible generalisations. Recently, generalisations such as to positive maps between matrix algebras have become more and more interesting for applications. This text gives a review of over 70 years of matrix scaling. The focus lies on the mathematical landscape surrounding the problem and its solution as well as the generalisation to positive maps and contains hardly any nontrivial unpublished results.

An Expert Overview of Matrix Scaling and Sinkhorn's Theorem

The paper "A review of matrix scaling and Sinkhorn's normal form for matrices and positive maps" by Martin Idel offers a detailed examination of the mathematical theory surrounding matrix scaling, particularly the results associated with Sinkhorn's theorem. This theorem provides a method for transforming a nonnegative matrix into a doubly stochastic matrix (where each row and column sums to one) by appropriate diagonal scaling. The paper not only reviews the classical results but also explores generalizations to positive maps between matrix algebras, which have grown in prominence for their applications in quantum mechanics and other fields.

Overview of Matrix Scaling

At the heart of the paper lies the matrix scaling problem: for a given nonnegative matrix $A$, find diagonal matrices $D_1$ and $D_2$ such that $D_1 A D_2$ is doubly stochastic. Sinkhorn's theorem guarantees that such matrices $D_1$ and $D_2$ exist under appropriate conditions. The paper explores numerous approaches for proving Sinkhorn's theorem, including potential methods, nonlinear Perron-Frobenius theory, entropy optimization, and convex programming. Each of these approaches offers unique insights and methodologies for tackling the matrix scaling problem, providing a rich landscape of mathematical tools.

Key Numerical and Conceptual Contributions

One of the seminal contributions of the paper is its analysis of the convergence properties of the matrix scaling algorithms. In particular, it highlights the efficiency and numerical stability of the Sinkhorn-Knopp algorithm, which iteratively adjusts the rows and columns of the matrix to achieve the desired doubly stochastic form. The paper also effectively addresses the generalizations of matrix scaling to scenarios involving positive maps, notably in the context of quantum mechanics, where the doubly stochastic maps have particular physical interpretations.

Generalizations to Positive Maps

In extending the matrix scaling problem to positive maps, the paper tackles a more abstract setting wherein the entries of matrices are replaced by operators acting on a Hilbert space. This generalization is significant in the paper of quantum mechanics, where positive maps represent quantum channels that must be normalized similarly to doubly stochastic matrices. The paper provides rigorous conditions under which these positive maps can be scaled, maintaining the structure required for various applications.

Future Developments and Open Problems

The paper speculates on future developments, particularly the potential application of matrix scaling techniques to other areas such as optimisation of quantum operations or the design of new quantum algorithms. It opens a discussion on the computational complexity associated with scaling large matrices and positive maps, prompting further research into efficient algorithmic implementations that can leverage modern computational resources.

Overall, Idel's paper is a comprehensive review that not only elucidates the theoretical underpinnings of matrix scaling and Sinkhorn's theorem but also propels the discourse into modern applications, particularly in the field of quantum information. It stands as a valuable resource for researchers aiming to extend these mathematical tools to increasingly complex and multidimensional applications across disciplines.