Proofs of Central Limit Error Scaling Lemmas for ESPRIT

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

C Deferred proofs for Section 2

In this section, we prove Lemmas 2.1 and 2.2 from Section 2. We begin in Appendix C.1 by proving some supplementary error bounds that we will use. Then, we prove Lemma 2.1 in Appendix C.2 and Lemma 2.2 in Appendix C.3.

C.1 Error matrix bounds

This result appears in the proof of [Mec07, Thm. 2 (Page 320)] for Ej real. Estimates of this form with explicit constants in the real or complex case can easily be obtained from the theory of matrix concentration (see, e.g., [Tro15, sec. 4.4] for a specific calculation).

Using this result, we can bound the norm of the error matrix in total:

Proposition C.2 (Error matrix, norm bound). It holds that

Proposition C.3 (Tail error in the dominant eigenspace). We have the bound

C.2 Proof of Lemma 2.1

In this section, we provide a proof of Lemma 2.1, broken into two steps.

Step 1: Proof of Eq. (2.1). The result of Eq. (2.1) follows directly by combining matrix perturbation theorems (Appendix A.5), singular value bounds for Vandermonde matrices (Appendix B.1), and the error matrix bounds developed in Appendix C.1.

By Weyl’s theorem (Theorem A.7) and Proposition C.2,

By Ostrowski’s theorem (Theorem A.8) and Theorem B.1,

C.3 Proof of Lemma 2.2

The proof of the lemma is then completed.