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Approximations of Extremal Eigenspace and Orthonormal Polar Factor

Ren-Cang Li

TL;DR

This work addresses two extremal trace-based problems: (i) bounding the error in approximating the top-$k$ eigenspace of a Hermitian matrix $H$ via the difference between $ ext{tr}(P_*^HHP_*)$ and $ ext{tr}(P^HP)$, and (ii) bounding the error in the orthonormal polar factor of a matrix $B$ by approximating its nuclear norm with $ ext{tr}(P^HB)$. The authors derive tight bounds in terms of canonical subspace distances, introducing parameters $ ext{η}$ and $ ext{ε}$ to quantify the deviation and connecting these to subspace angles via $oldsymbol{ ext{ε}}=ig( ext{η}/( ext{λ}_k(H)- ext{λ}_{k+1}(H))ig)^{1/2}$. A key contribution is the complex-field extension of known real-field results, including a new inequality for the top eigenspace error and a comprehensive set of bounds for polar-factor recovery using the nuclear norm. These results underpin convergence analyses for Stiefel-manifold optimization methods used in data-science tasks and provide rigorous guarantees for subspace and polar-factor approximations in applications requiring trace-norm or nuclear-norm considerations.

Abstract

This paper is concerned with two extremal problems from matrix analysis. One is about approximating the top eigenspaces of a Hermitian matrix and the other one about approximating the orthonormal polar factor of a general matrix. Tight error bounds on the quality of the approximations are obtained.

Approximations of Extremal Eigenspace and Orthonormal Polar Factor

TL;DR

This work addresses two extremal trace-based problems: (i) bounding the error in approximating the top- eigenspace of a Hermitian matrix via the difference between and , and (ii) bounding the error in the orthonormal polar factor of a matrix by approximating its nuclear norm with . The authors derive tight bounds in terms of canonical subspace distances, introducing parameters and to quantify the deviation and connecting these to subspace angles via . A key contribution is the complex-field extension of known real-field results, including a new inequality for the top eigenspace error and a comprehensive set of bounds for polar-factor recovery using the nuclear norm. These results underpin convergence analyses for Stiefel-manifold optimization methods used in data-science tasks and provide rigorous guarantees for subspace and polar-factor approximations in applications requiring trace-norm or nuclear-norm considerations.

Abstract

This paper is concerned with two extremal problems from matrix analysis. One is about approximating the top eigenspaces of a Hermitian matrix and the other one about approximating the orthonormal polar factor of a general matrix. Tight error bounds on the quality of the approximations are obtained.
Paper Structure (5 sections, 5 theorems, 40 equations)

This paper contains 5 sections, 5 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Canonical angles between subspaces
  3. Extremal Eigenspace
  4. Orthonormal Polar Factor
  5. Concluding Remarks

Key Result

Theorem 3.1

Let $H\in\mathbb{C}^{n\times n}$ be Hermitian and $P_*\in{\rm St}(k,n)$ whose column space ${\cal R}(P_*)$ is the invariant subspace of $H$ associated with its $k$ largest eigenvalues. Suppose that $\lambda_k(H)-\lambda_{k+1}(H)>0$. Given $P\in{\rm St}(k,n)$, let ThenThe first inequality in eq:eig2max actually holds so long as ${\cal R}(P_*)$ is a $k$-dimensional invariant subspace of $H$, as its

Theorems & Definitions (8)

  • Theorem 3.1
  • proof
  • Lemma 4.1: von Neumann's trace inequality neum:1937, hojo:1991
  • Lemma 4.2
  • proof
  • Theorem 4.1
  • proof
  • Corollary 4.1