TY - GEN

T1 - A quasi-random approach to matrix spectral analysis

AU - Ben-Or, Michael

AU - Eldar, Lior

N1 - Publisher Copyright: © Michael Ben-Or and Lior Eldar.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Inspired by quantum computing algorithms for Linear Algebra problems [6, 14] we study how simulation on a classical computer of this type of “Phase Estimation algorithms” performs when we apply it to the Eigen-Problem of Hermitian matrices. The result is a completely new, efficient and stable, parallel algorithm to compute an approximate spectral decomposition of any Hermitian matrix. The algorithm can be implemented by Boolean circuits in O(log2n) parallel time with a total cost of O(nω+1) Boolean operations. This Boolean complexity matches the best known O(log2n) parallel time algorithms, but unlike those algorithms our algorithm is (logarithmically) stable, so it may lead to actual implementations, allowing fast parallel computation of eigenvectors and eigenvalues in practice. Previous approaches to solve the Eigen-Problem generally use randomization to avoid bad conditions - as we do. Our algorithm makes further use of randomization in a completely new way, taking random powers of a unitary matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian perturbation and a random polynomial power are su cient to ensure almost pairwise independence of the phases (mod 2π) is the main technical contribution of this work. It relies on the theory of low-discrepancy or quasi-random sequences - a theory, which to the best of our knowledge, has not been connected thus far to linear algebra problems. Hence, we believe that further study of this new connection will lead to additional improvements.

AB - Inspired by quantum computing algorithms for Linear Algebra problems [6, 14] we study how simulation on a classical computer of this type of “Phase Estimation algorithms” performs when we apply it to the Eigen-Problem of Hermitian matrices. The result is a completely new, efficient and stable, parallel algorithm to compute an approximate spectral decomposition of any Hermitian matrix. The algorithm can be implemented by Boolean circuits in O(log2n) parallel time with a total cost of O(nω+1) Boolean operations. This Boolean complexity matches the best known O(log2n) parallel time algorithms, but unlike those algorithms our algorithm is (logarithmically) stable, so it may lead to actual implementations, allowing fast parallel computation of eigenvectors and eigenvalues in practice. Previous approaches to solve the Eigen-Problem generally use randomization to avoid bad conditions - as we do. Our algorithm makes further use of randomization in a completely new way, taking random powers of a unitary matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian perturbation and a random polynomial power are su cient to ensure almost pairwise independence of the phases (mod 2π) is the main technical contribution of this work. It relies on the theory of low-discrepancy or quasi-random sequences - a theory, which to the best of our knowledge, has not been connected thus far to linear algebra problems. Hence, we believe that further study of this new connection will lead to additional improvements.

KW - Eigenvalues

KW - Eigenvectors

KW - Low-discrepancy sequence

UR - http://www.scopus.com/inward/record.url?scp=85041678133&partnerID=8YFLogxK

U2 - https://doi.org/10.4230/LIPIcs.ITCS.2018.6

DO - https://doi.org/10.4230/LIPIcs.ITCS.2018.6

M3 - Conference contribution

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 9th Innovations in Theoretical Computer Science, ITCS 2018

A2 - Karlin, Anna R.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 9th Innovations in Theoretical Computer Science, ITCS 2018

Y2 - 11 January 2018 through 14 January 2018

ER -