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
B Vandermonde matrice
In this section, we discuss results for Vandermonde matrices that will be heavily used throughout our analysis. We begin in Appendix B.1 by discussing Moitra’s bounds [Moi15] for the singular values of a Vandermonde matrix. Then, in Appendix B.2, we present a comparison lemma for the Vandermonde basis and eigenbasis of a Toeplitz matrix
B.1 Moitra’s singular value bound
For us, one of the core technical tools for analyzing ESPRIT is bound on the singular values of Vandermonde matrices, proven in the seminal work of Moitra [Moi15]. The main result is as follow:
This result immediately yields the following useful corollary
Corollary B.2 (Vandermonde matrix: Gram matrix and outer product). In the setting of Theorem B.
B.2 Vandermonde basis and eigenbasis
The exact Toeplitz matrix T Eq. (1.6) has two useful factorizations, each of which packages useful information about the matrix. The first factorization is the eigendecomposition
The eigendecomposition is useful because it can be easily computed using standard methods from numerical linear algebra. The second useful factorization is the Vandermonde decomposition
The Vandermonde decomposition contains information about the frequency spectrum. However, it can only be computed indirectly, such as by procedures such as ESPRIT which use the eigendecomposition as a subroutine.
Recall that
A direct consequence of Theorem B.1 is the following estimates for the eigenvalues of T:
The lemma is then proved.
By the definition of the Vandermonde matrix,
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.