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<p>We investigate an algorithm for the least squares fitting of a subset of the eigenvalues of an unknown Hermitian matrix lying on an affine subspace, called the Lift and Projection (LP) method, due to Chen and Chu (SIAM Journal on Numerical Analysis, 33 (1996), pp.&#xa0;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 method 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<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\times \)</EquationSource> <EquationSource Format="MATHML"><math> <mo>×</mo> </math></EquationSource> </InlineEquation>32400 sparse matrix from a 5-dimensional subspace of spin Hamiltonian matrices.</p>

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A Riemannian gradient descent method for the least squares inverse eigenvalue problem

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月13日

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  • Alban Bloor Riley,
  • Marcus Webb,
  • Michael L. Baker

摘要

We investigate an algorithm for the least squares fitting of a subset of the eigenvalues of an unknown Hermitian matrix lying on 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 method 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 \(\times \) × 32400 sparse matrix from a 5-dimensional subspace of spin Hamiltonian matrices.