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 1.4 Technical overview In this section, we present an overview of the new technical advancements that are essential in our improved analysis. While these findings are specifically designed for the ESPRIT algorithm, we believe the underlying concepts may also apply to other subspace-based signal processing techniques. Therefore, we anticipate that these technical results may be of independent interest. 1.4.1 Vandermonde matrix and eigenbasis 1.4.2 Second-order perturbation for dominant eigenspace We believe that a result of the form Lemma 1.8 should also exist for non-Hermitian Hankel and Toeplitz matrices, allowing our methods to be extended to study the error scaling of other spectral estimation algorithms. 1.5 Organization The rest of the paper is structured as follows. In Section 2, we present the direct extension of the previous theoretical findings to prove Theorem 1.2 for completeness. The formal statement and the proof of Theorem 1.4 are provided in Section 3, under the assumption of the correctness of Lemma 4.1 and Theorem 1.9. Section 4 is dedicated to discussing the proof of Lemma 4.1, while in Section 5, we provide the proof of Theorem 1.9. In Appendix A, we define the notations and provide some preliminaries. In Appendix B, we establish some properties of Vandermonde matrix. Appendices C to E contain the technical details and proofs of Sections 2, 4 and 5, respectively. In Appendix F, we prove the lower bound for the spectral estimation problem. This paper is available on arxiv under CC BY 4.0 DEED license. [5] Here, and going forward, all projectors are orthogonal projectors. 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. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 ESPRIT algorithm and central limit error scaling 1.1 ESPRIT algorithm and central limit error scaling 1.2 Contribution 1.2 Contribution 1.3 Related work 1.3 Related work 1.4 Technical overview and 1.5 Organization 1.4 Technical overview and 1.5 Organization 2 Proof of the central limit error scaling 2 Proof of the central limit error scaling 3 Proof of the optimal error scaling 3 Proof of the optimal error scaling 4 Second-order eigenvector perturbation theory 4 Second-order eigenvector perturbation theory 5 Strong eigenvector comparison 5 Strong eigenvector comparison 5.1 Construction of the “good” P 5.1 Construction of the “good” P 5.2 Taylor expansion with respect to the error terms 5.2 Taylor expansion with respect to the error terms 5.3 Error cancellation in the Taylor expansion 5.3 Error cancellation in the Taylor expansion 5.4 Proof of Theorem 5.1 5.4 Proof of Theorem 5.1 A Preliminaries A Preliminaries B Vandermonde matrice B Vandermonde matrice C Deferred proofs for Section 2 C Deferred proofs for Section 2 D Deferred proofs for Section 4 D Deferred proofs for Section 4 E Deferred proofs for Section 5 E Deferred proofs for Section 5 F Lower bound for spectral estimation F Lower bound for spectral estimation References References 1.4 Technical overview In this section, we present an overview of the new technical advancements that are essential in our improved analysis. While these findings are specifically designed for the ESPRIT algorithm, we believe the underlying concepts may also apply to other subspace-based signal processing techniques. Therefore, we anticipate that these technical results may be of independent interest. 1.4.1 Vandermonde matrix and eigenbasis 1.4.1 Vandermonde matrix and eigenbasis 1.4.2 Second-order perturbation for dominant eigenspace 1.4.2 Second-order perturbation for dominant eigenspace We believe that a result of the form Lemma 1.8 should also exist for non-Hermitian Hankel and Toeplitz matrices, allowing our methods to be extended to study the error scaling of other spectral estimation algorithms. 1.5 Organization The rest of the paper is structured as follows. In Section 2, we present the direct extension of the previous theoretical findings to prove Theorem 1.2 for completeness. The formal statement and the proof of Theorem 1.4 are provided in Section 3, under the assumption of the correctness of Lemma 4.1 and Theorem 1.9. Section 4 is dedicated to discussing the proof of Lemma 4.1, while in Section 5, we provide the proof of Theorem 1.9. In Appendix A, we define the notations and provide some preliminaries. In Appendix B, we establish some properties of Vandermonde matrix. Appendices C to E contain the technical details and proofs of Sections 2, 4 and 5, respectively. In Appendix F, we prove the lower bound for the spectral estimation problem. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv [5] Here, and going forward, all projectors are orthogonal projectors. 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. Authors: 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.

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