Second-Order Eigenvector Perturbation Theory for ESPRIT Analysis

Table of Links

Abstract and 1 Introduction

1.1 ESPRIT algorithm and central limit error scaling

1.2 Contribution

1.3 Related work

1.4 Technical overview and 1.5 Organization

2 Proof of the central limit error scaling

3 Proof of the optimal error scaling

4 Second-order eigenvector perturbation theory

5 Strong eigenvector comparison

5.1 Construction of the “good” P

5.2 Taylor expansion with respect to the error terms

5.3 Error cancellation in the Taylor expansion

5.4 Proof of Theorem 5.1

A Preliminaries

B Vandermonde matrice

C Deferred proofs for Section 2

D Deferred proofs for Section 4

E Deferred proofs for Section 5

F Lower bound for spectral estimation

References

4 Second-order eigenvector perturbation theory

In this section, we provide a formal statement of the second-order eigenvector perturbation result in Lemma 1.8. We outline the main steps involved in proving this lemma and defer technical details to Appendix D. In the following part of this section, we always assume Eq. (1.3), Eq. (1.4), and Theorem 3.1 Eq. (3.1) and omit these conditions later.

Proposition 4.2 (Expansion of spectral projector). It holds that

The series expansion in Proposition 4.2 is rather nontransparent, as the higher-order terms are expressed in terms of contour integrals. Fortunately, these integrals can be evaluated exactly, leading to explicit formulas for all terms in the expansion in terms of Schur polynomials. As the results are rather intricate, so we defer the statement and proof to Appendix D.1. We can then bound the higher-order terms in this expression, yielding the following representation:

Lemma 4.3 (Expansion of spectral projector, simplified). It holds that

Thus, we can write

where R is the remainder matrix of norm

The RHS becomes

Comparing LHS and RHS with Eq. (4.9), we see that

Now, we estimate

Summing over k, we get that

The proof of the lemma is then completed.