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A Riemannian Gradient Descent Method for the Least Squares Inverse Eigenvalue Problem

Alban Bloor Riley, Marcus Webb, Michael L. Baker

TL;DR

The paper reinterprets the Lift and Projection method for the Least Squares Inverse Eigenvalue Problem (LSIEP) as a Riemannian Gradient Descent (RGD) on the parameter space with a metric induced by the affine map $A(x)$. This perspective yields a cheaper update based on a partial eigendecomposition and a formal descent guarantee, linked through the Gram matrix $B$ with the Hessian term $H_F$. The authors prove equivalence between LP and a gradient-step scheme $x^{k+1}=x^k - B^{-1}\nabla F$, and establish global convergence with a quantitative decrease bound, even allowing a doubled-step variant. Numerical experiments on Toeplitz LSIEP and large spin Hamiltonian problems from Inelastic Neutron Scattering demonstrate substantial speedups, particularly when the problem is small in the spectral subspace or when eigenvalues are extremal or near a target, validating the practical impact of the RGDLp reformulation.

Abstract

We address an algorithm for the least squares fitting of a subset of the eigenvalues of an unknown Hermitian matrix lying an an affine subspace, called the Lift and Projection (LP) method, due to Chen and Chu (SIAM Journal on Numerical Analysis, 33 (1996), pp.2417-2430). The LP method iteratively `lifts' the current iterate onto the spectral constraint manifold then 'projects' onto the solution's affine subspace. We prove that this is equivalent to a Riemannian Gradient Descent with respect to a natural Riemannian metric. This insight allows us to derive a more efficient implementation, analyse more precisely its global convergence properties, and naturally append additional constraints to the problem. We provide several numerical experiments to demonstrate the improvement in computation time, which can be more than an order of magnitude if the eigenvalue constraints are on the smallest eigenvalues, the largest eigenvalues, or the eigenvalues closest to a given number. These experiments include an inverse eigenvalue problem arising in Inelastic Neutron Scattering of Manganese-6, which requires the least squares fitting of 16 experimentally observed eigenvalues of a $32400\times32400$ sparse matrix from a 5-dimensional subspace of spin Hamiltonian matrices.

A Riemannian Gradient Descent Method for the Least Squares Inverse Eigenvalue Problem

TL;DR

The paper reinterprets the Lift and Projection method for the Least Squares Inverse Eigenvalue Problem (LSIEP) as a Riemannian Gradient Descent (RGD) on the parameter space with a metric induced by the affine map . This perspective yields a cheaper update based on a partial eigendecomposition and a formal descent guarantee, linked through the Gram matrix with the Hessian term . The authors prove equivalence between LP and a gradient-step scheme , and establish global convergence with a quantitative decrease bound, even allowing a doubled-step variant. Numerical experiments on Toeplitz LSIEP and large spin Hamiltonian problems from Inelastic Neutron Scattering demonstrate substantial speedups, particularly when the problem is small in the spectral subspace or when eigenvalues are extremal or near a target, validating the practical impact of the RGDLp reformulation.

Abstract

We address an algorithm for the least squares fitting of a subset of the eigenvalues of an unknown Hermitian matrix lying an an affine subspace, called the Lift and Projection (LP) method, due to Chen and Chu (SIAM Journal on Numerical Analysis, 33 (1996), pp.2417-2430). The LP method iteratively `lifts' the current iterate onto the spectral constraint manifold then 'projects' onto the solution's affine subspace. We prove that this is equivalent to a Riemannian Gradient Descent with respect to a natural Riemannian metric. This insight allows us to derive a more efficient implementation, analyse more precisely its global convergence properties, and naturally append additional constraints to the problem. We provide several numerical experiments to demonstrate the improvement in computation time, which can be more than an order of magnitude if the eigenvalue constraints are on the smallest eigenvalues, the largest eigenvalues, or the eigenvalues closest to a given number. These experiments include an inverse eigenvalue problem arising in Inelastic Neutron Scattering of Manganese-6, which requires the least squares fitting of 16 experimentally observed eigenvalues of a sparse matrix from a 5-dimensional subspace of spin Hamiltonian matrices.

Paper Structure

This paper contains 11 sections, 8 theorems, 51 equations, 2 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Lift and Projection
  3. Lift
  4. Projection
  5. Algorithm Equivalence
  6. Riemannian Gradient Descent
  7. Riemannian Geometry
  8. The Lift and Projection method as a Riemannian gradient descent method
  9. Global Convergence
  10. Numerical Experiments
  11. Conclusion

Key Result

Theorem 1

\newlabelthrm. LP is RGD0 The lift and projection method is equivalent to the Riemannian gradient descent method on $\mathbb R^\ell$ equipped with the metric induced by $A(x)$ (in the sense of Definition def. Induced Metric).

Figures (2)

  • Figure 1: Spyplots showing the sparsity of $A_1,A_2$ and $A_3$ for Example \ref{['Ex. Cr6']}
  • Figure 2: Convergence plot of the RGD LP Min method for Example \ref{['Ex. Mn6']}

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Theorem 1: Theorem 4.1 from chen_least_1996
  • Theorem 2
  • Lemma 3
  • Proof 1
  • Proof 2: Proof of Theorem \ref{['Thrm. Alg equivalence']}
  • Definition 1: boumal_introduction_2023
  • Definition 2: boumal_introduction_2023
  • Definition 3: boumal_introduction_2023
  • ...and 15 more