How to project onto SL($n$)

TL;DR

This paper studies the problem of projecting matrices onto the special linear group SL(n) using Frobenius norm, revealing that the projection can be simplified to diagonal matrices, and introduces efficient algorithms with proven convergence.

Contribution

It demonstrates that the projection onto SL(n) can be reduced to diagonal matrices and proposes four convergent iterative algorithms for this task.

Findings

Projection reduces to diagonal matrices

Algorithms converge reliably

Computational cost comparable to SVD

Abstract

We consider the closest-point projection with respect to the Frobenius norm of a general real square matrix to the set SL($n$) of matrices with unit determinant. As it turns out, it is sufficient to consider diagonal matrices only. We investigate the structure of the problem both in Euclidean coordinates and in an $n$-dimensional generalization of the classical hyperbolic coordinates of the positive quadrant. Using symmetry arguments we show that the global minimizer is contained in a particular cone. Based on different views of the problem, we propose four different iterative algorithms, and we give convergence results for all of them. Numerical tests show that computing the projection costs essentially as much as a singular value decomposition. Finally, we give an explicit formula for the first derivative of the projection.