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The condition number of singular subspaces, revisited

Nick Vannieuwenhoven

TL;DR

This work provides a clear, expository derivation of the Rice-type condition number for computing left (and hence right) singular subspaces, presenting an alternative computation in terms of the Euclidean input-distance and unitarily invariant Grassmannian distances. For real matrices with distinct nonzero singular values, the condition number κ_π(A) is given by the maximum inverse singular-value-gap over i∈π and j∈π^c, scaled by a factor between 1/√2 and 1, namely κ_π(A) = max_{i∈π,j∈π^c} [1/|σ_i−σ_j|] sqrt{(σ_i^2+σ_j^2)/(σ_i+σ_j)^2}; the result extends to the cokernel case and to complex matrices via a restriction-of-scalars argument. The paper also proves symmetry between left and right subspaces, invariance under field extension, a first-order sharp perturbation bound, and identifies a worst-case perturbation direction achieving the bound. A concrete real-data example illustrates well- and ill-conditioned cases and confirms the theoretical predictions. Overall, the work links geometric perturbation theory on Grassmannians with a compact, explicit condition-number formula for singular subspaces, bridging classical perturbation theory and modern manifold-valued analysis.

Abstract

I revisit the condition number of computing left and right singular subspaces from [J.-G. Sun, Perturbation analysis of singular subspaces and deflating subspaces, Numer. Math. 73(2), pp. 235--263, 1996]. For real and complex matrices, I present an alternative computation of this condition number in the Euclidean distance on the input space of matrices and the chordal, Grassmann, and Procrustes distances on the output Grassmannian manifold of linear subspaces. Up to a small factor, this condition number equals the inverse minimum singular value gap between the singular values corresponding to the selected singular subspace and those not selected.

The condition number of singular subspaces, revisited

TL;DR

This work provides a clear, expository derivation of the Rice-type condition number for computing left (and hence right) singular subspaces, presenting an alternative computation in terms of the Euclidean input-distance and unitarily invariant Grassmannian distances. For real matrices with distinct nonzero singular values, the condition number κ_π(A) is given by the maximum inverse singular-value-gap over i∈π and j∈π^c, scaled by a factor between 1/√2 and 1, namely κ_π(A) = max_{i∈π,j∈π^c} [1/|σ_i−σ_j|] sqrt{(σ_i^2+σ_j^2)/(σ_i+σ_j)^2}; the result extends to the cokernel case and to complex matrices via a restriction-of-scalars argument. The paper also proves symmetry between left and right subspaces, invariance under field extension, a first-order sharp perturbation bound, and identifies a worst-case perturbation direction achieving the bound. A concrete real-data example illustrates well- and ill-conditioned cases and confirms the theoretical predictions. Overall, the work links geometric perturbation theory on Grassmannians with a compact, explicit condition-number formula for singular subspaces, bridging classical perturbation theory and modern manifold-valued analysis.

Abstract

I revisit the condition number of computing left and right singular subspaces from [J.-G. Sun, Perturbation analysis of singular subspaces and deflating subspaces, Numer. Math. 73(2), pp. 235--263, 1996]. For real and complex matrices, I present an alternative computation of this condition number in the Euclidean distance on the input space of matrices and the chordal, Grassmann, and Procrustes distances on the output Grassmannian manifold of linear subspaces. Up to a small factor, this condition number equals the inverse minimum singular value gap between the singular values corresponding to the selected singular subspace and those not selected.
Paper Structure (14 sections, 13 theorems, 76 equations)

This paper contains 14 sections, 13 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Statement of the results
  3. Illustration in an example
  4. Proof of \ref{['prop_only_left', 'prop_complementary']}
  5. Proof of \ref{['thm_main']}: The real case
  6. Simplifying assumptions
  7. From a manifold to the tangent space
  8. Characterizing $\mathcal{L}_\pi^\mathbb{R}$
  9. Simplifying norms
  10. Eliminating variables
  11. Computing the condition number
  12. Continuous extension to all real matrices
  13. Proof of \ref{['thm_main']}: The complex case
  14. Proof of \ref{['thm_secondary']}

Key Result

Lemma 1

If we select the chordal, Grassmann, and Procrustes distances from eqn_distance on $\mathrm{Gr}(\ell,\Bbbk^n)$ and $\mathrm{Gr}(k,\Bbbk^m)$, and the Euclidean distance eqn_eucl_metric on $\Bbbk^{m\times n}$ and $\Bbbk^{n\times m}$, then the following equality of condition numbers of left and right s

Theorems & Definitions (22)

  • Lemma 1: Reduction to left singular subspaces
  • Theorem 1: Condition of left singular subspaces
  • Remark 1
  • Corollary 1: Error bound
  • Lemma 2: Condition of the complementary subspace
  • Corollary 2: Continuity
  • Corollary 3: Invariance under field extension
  • Proposition 1: Worst perturbation direction
  • proof : Proof of \ref{['prop_only_left']}
  • proof : Proof of \ref{['prop_complementary']}
  • ...and 12 more