Table of Links
1.1 ESPRIT algorithm and central limit error scaling
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
C Deferred proofs for Section 2
D Deferred proofs for Section 4
E Deferred proofs for Section 5
F Lower bound for spectral estimation
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.
This paper is available on arxiv under CC BY 4.0 DEED license.
Authors:
(1) Zhiyan Ding, Department of Mathematics, University of California, Berkeley;
(2) Ethan N. Epperly, Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA;
(3) Lin Lin, Department of Mathematics, University of California, Berkeley, Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, and Challenge Institute for Quantum Computation, University of California, Berkeley;
(4) Ruizhe Zhang, Simons Institute for the Theory of Computing.
