TY - GEN

T1 - Sublinear time spectral density estimation

AU - Braverman, Vladimir

AU - Krishnan, Aditya

AU - Musco, Christopher

N1 - Funding Information:
We thank Cameron Musco, Raphael Meyer, and Tyler Chen for helpful discussions and suggestions. This research was supported in part by NSF CAREER grants 2045590 and 1652257, ONR Award N00014-18-1-2364, and the Lifelong Learning Machines program from DARPA/MTO.
Publisher Copyright:
© 2022 ACM.

PY - 2022/9/6

Y1 - 2022/9/6

N2 - We present a new sublinear time algorithm for approximating the spectral density (eigenvalue distribution) of an n× n normalized graph adjacency or Laplacian matrix. The algorithm recovers the spectrum up to "accuracy in the Wasserstein-1 distance in O(n· (1/")) time given sample access to the graph. This result compliments recent work by David Cohen-Steiner, Weihao Kong, Christian Sohler, and Gregory Valiant (2018), which obtains a solution with runtime independent of n, but exponential in 1/". We conjecture that the trade-off between dimension dependence and accuracy is inherent. Our method is simple and works well experimentally. It is based on a Chebyshev polynomial moment matching method that employees randomized estimators for the matrix trace. We prove that, for any Hermitian A, this moment matching method returns an "approximation to the spectral density using just O(1/") matrix-vector products with A. By leveraging stability properties of the Chebyshev polynomial three-term recurrence, we then prove that the method is amenable to the use of coarse approximate matrix-vector products. Our sublinear time algorithm follows from combining this result with a novel sampling algorithm for approximating matrix-vector products with a normalized graph adjacency matrix. Of independent interest, we show a similar result for the widely used kernel polynomial method (KPM), proving that this practical algorithm nearly matches the theoretical guarantees of our moment matching method. Our analysis uses tools from Jackson's seminal work on approximation with positive polynomial kernels.

AB - We present a new sublinear time algorithm for approximating the spectral density (eigenvalue distribution) of an n× n normalized graph adjacency or Laplacian matrix. The algorithm recovers the spectrum up to "accuracy in the Wasserstein-1 distance in O(n· (1/")) time given sample access to the graph. This result compliments recent work by David Cohen-Steiner, Weihao Kong, Christian Sohler, and Gregory Valiant (2018), which obtains a solution with runtime independent of n, but exponential in 1/". We conjecture that the trade-off between dimension dependence and accuracy is inherent. Our method is simple and works well experimentally. It is based on a Chebyshev polynomial moment matching method that employees randomized estimators for the matrix trace. We prove that, for any Hermitian A, this moment matching method returns an "approximation to the spectral density using just O(1/") matrix-vector products with A. By leveraging stability properties of the Chebyshev polynomial three-term recurrence, we then prove that the method is amenable to the use of coarse approximate matrix-vector products. Our sublinear time algorithm follows from combining this result with a novel sampling algorithm for approximating matrix-vector products with a normalized graph adjacency matrix. Of independent interest, we show a similar result for the widely used kernel polynomial method (KPM), proving that this practical algorithm nearly matches the theoretical guarantees of our moment matching method. Our analysis uses tools from Jackson's seminal work on approximation with positive polynomial kernels.

KW - approximate matrix-vector multiplication

KW - density estimation

KW - graph approximation

KW - moment matching

UR - http://www.scopus.com/inward/record.url?scp=85132184321&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85132184321&partnerID=8YFLogxK

U2 - 10.1145/3519935.3520009

DO - 10.1145/3519935.3520009

M3 - Conference contribution

AN - SCOPUS:85132184321

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 1144

EP - 1157

BT - STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

A2 - Leonardi, Stefano

A2 - Gupta, Anupam

PB - Association for Computing Machinery

T2 - 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022

Y2 - 20 June 2022 through 24 June 2022

ER -