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How to project onto SL($n$)

Patrick Jaap, Oliver Sander

TL;DR

This work shows that projecting a real matrix onto the special linear group $\textup{SL}(n)$ under the Frobenius norm reduces to a diagonal problem due to unitary invariance. It develops four complementary algorithms leveraging diagonalization, logarithmic and hyperbolic coordinates, and constrained/unnconstrained Newton methods, with convergence guarantees and practical performance analyses. A detailed study of the admissible set’s geometry reveals when the minimizer is unique and how multiple stationary points can arise, especially for $n\ge3$, and the authors provide a constructive solution path and an explicit derivative of the projection. The results have direct relevance to finite-strain elastoplasticity and projection-based discretizations, where efficient and differentiable SL$(n)$ projections at quadrature points are essential.

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.

How to project onto SL($n$)

TL;DR

This work shows that projecting a real matrix onto the special linear group under the Frobenius norm reduces to a diagonal problem due to unitary invariance. It develops four complementary algorithms leveraging diagonalization, logarithmic and hyperbolic coordinates, and constrained/unnconstrained Newton methods, with convergence guarantees and practical performance analyses. A detailed study of the admissible set’s geometry reveals when the minimizer is unique and how multiple stationary points can arise, especially for , and the authors provide a constructive solution path and an explicit derivative of the projection. The results have direct relevance to finite-strain elastoplasticity and projection-based discretizations, where efficient and differentiable SL projections at quadrature points are essential.

Abstract

We consider the closest-point projection with respect to the Frobenius norm of a general real square matrix to the set SL() 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 -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.

Paper Structure

This paper contains 26 sections, 26 theorems, 141 equations, 13 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Transformation to diagonal matrices
  3. The structure of the admissible set, and the global minimizer
  4. The minimization problem in $\mathbb R^n$
  5. The positive orthant
  6. The order cone
  7. Uniqueness and non-uniqueness in the order cone
  8. Uniqueness of the global minimizer in the order cone for $n=2$
  9. The case $n \ge 3$
  10. Coordinate transformations
  11. Logarithmic coordinates
  12. The squared Euclidean distance in logarithmic coordinates
  13. Hyperbolic coordinates
  14. The squared Euclidean distance in hyperbolic coordinates
  15. Algorithms that compute the projection
  16. ...and 11 more sections

Key Result

Lemma 3.1

Let $a \in \mathbb R^n_{\ge 0}$. Then there is a global minimizer $p \in \sl(n)$ with $p_i > 0$ for all $i\in\{1,\hdots,n\}$.

Figures (13)

  • Figure 1: The sets $\sl(2)$ (left) and $\sl(3)$ (right). The colored lines (left) and surfaces (right) show the actual sets $\sl(n)$. The grey area on the left is the set $\sl^+(n) \cap \mathbb R^n_{\ge 0}$ whose strict convexity is proved in Lemma \ref{['lemma:G+-is-convex']}. In the right picture, it is the set beyond the blue surface.
  • Figure 2: The order cones for $n=2$ and $n=3$
  • Figure 3: Illustrations of the proofs of Lemmas \ref{['lem:uniqueness-n-2']} and \ref{['lem:non_generic_case_2d']}, which both discuss the case $n=2$. Left: If two points $p$ and $q$ in the interior of the order cone (in grey) are both stationary for the squared distance to a point $a$, then this point $a$ cannot also be in the order cone. Right: If $a_1 = a_2 > 2$, then there are exactly three stationary points.
  • Figure 4: The solution path for $a=(2.5, 2)^T$
  • Figure 5: The expression $\operatorname{prod}(\mathcal{P}(\lambda))$ as a function of $\lambda$, for $a=(1.92, 1.9199, 1.9198)^T$
  • ...and 8 more figures

Theorems & Definitions (56)

  • Remark 1.1: Radial scaling
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Lemma 3.5
  • proof
  • ...and 46 more