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Riemannian optimization on the symplectic Stiefel manifold using second-order information

Rasmus Jensen, Ralf Zimmermann

TL;DR

A matrix formula for the Riemannian Hessian under a right-invariant metric is derived and a novel retraction for approximating the Riemannian geodesics is proposed for approximating the Riemannian geodesics.

Abstract

Riemannian optimization is concerned with problems, where the independent variable lies on a smooth manifold. There is a number of problems from numerical linear algebra that fall into this category, where the manifold is usually specified by special matrix structures, such as orthogonality or definiteness. Following this line of research, we investigate tools for Riemannian optimization on the symplectic Stiefel manifold. We complement the existing set of numerical optimization algorithms with a Riemannian trust region method tailored to the symplectic Stiefel manifold. To this end, we derive a matrix formula for the Riemannian Hessian under a right-invariant metric. Moreover, we propose a novel retraction for approximating the Riemannian geodesics. Finally, we conduct a comparative study in which we juxtapose the performance of the Riemannian variants of the steepest descent, conjugate gradients, and trust region methods on selected matrix optimization problems that feature symplectic constraints.

Riemannian optimization on the symplectic Stiefel manifold using second-order information

TL;DR

A matrix formula for the Riemannian Hessian under a right-invariant metric is derived and a novel retraction for approximating the Riemannian geodesics is proposed for approximating the Riemannian geodesics.

Abstract

Riemannian optimization is concerned with problems, where the independent variable lies on a smooth manifold. There is a number of problems from numerical linear algebra that fall into this category, where the manifold is usually specified by special matrix structures, such as orthogonality or definiteness. Following this line of research, we investigate tools for Riemannian optimization on the symplectic Stiefel manifold. We complement the existing set of numerical optimization algorithms with a Riemannian trust region method tailored to the symplectic Stiefel manifold. To this end, we derive a matrix formula for the Riemannian Hessian under a right-invariant metric. Moreover, we propose a novel retraction for approximating the Riemannian geodesics. Finally, we conduct a comparative study in which we juxtapose the performance of the Riemannian variants of the steepest descent, conjugate gradients, and trust region methods on selected matrix optimization problems that feature symplectic constraints.
Paper Structure (24 sections, 7 theorems, 77 equations, 2 figures, 4 tables, 3 algorithms)

This paper contains 24 sections, 7 theorems, 77 equations, 2 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Original contributions
  3. State of the art
  4. Organization
  5. The symplectic Stiefel manifold
  6. The symplectic Stiefel manifold as a quotient space
  7. Normal space and projections
  8. Riemannian gradient of a scalar function on SpSt(2n,2k)
  9. Geodesics and retractions
  10. Vector transport on SpSt(2n,2k)
  11. The Riemannian Hessian of a scalar function on SpSt(2n,2k)
  12. Approximation of the Riemannian Hessian
  13. Optimzation algorithms on Riemannian manifolds
  14. Riemannian steepest descend and nonlinear conjugate gradients methods
  15. Riemannian trust--region method
  16. ...and 9 more sections

Key Result

Lemma 2.1

For the symplectic group $\textnormal{Sp}(2n)$ as an embedded submanifold of $\mathbb{R}^{2n\times 2n}$ endowed with the right invariant metric eq:rie_metric, the normal space, i.e., the orthogonal complement of the tangent space at $M\in \textnormal{Sp}(2n)$, is Its dimension is $(2n-1)n$.

Figures (2)

  • Figure 1: Numerical experiment with random point $X\in \mathop{\mathrm{\textnormal{SpSt}}}\nolimits(2n,2k)$ and random $\Delta\in \mathop{\mathrm{\textnormal{T}}}\nolimits_X\mathop{\mathrm{\textnormal{SpSt}}}\nolimits(2n,2k)$. Here $n = 100$ and $k = 10$. Left: Feasibility quantified by $\|\gamma(t)^+\gamma(t)-I_{2k}\|_F$ for the three different methods of moving on $\mathop{\mathrm{\textnormal{SpSt}}}\nolimits(2n,2k)$. The geodesic is computed according to \ref{['eq:geod_spst']}, Cayley 1 is \ref{['eq:simple_ret']} and Cayley 2 is \ref{['eq:eff_ret']}. Clearly, the computed geodesic does not remain on the manifold for large $t$. Right: The error measured in the Frobenious norm of the difference between the geodesic and the two retractions respectively.
  • Figure 2: Left: Continuation of the numerical experiment in Figure \ref{['fig:num_exp_geod_ret']} on $\mathop{\mathrm{\textnormal{SpSt}}}\nolimits(2n,2k), n=100,k=10$. Zooming in on the interval $[0.01,1]$ for $t$, both Caley retractions stay on the symplectic Stiefel manifold up to numerical errors of the order $10^{-15}$. Right: The retraction \ref{['eq:eff_ret']} is approximating the true geodesic up to an error that is several orders of magnitude lower than that of \ref{['eq:simple_ret']}.

Theorems & Definitions (8)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proof 1
  • Lemma 2.5
  • Theorem A.1: Levi--Civita connection
  • Lemma C.1