Sublinear Time Spectral Density Estimation

TL;DR

This paper introduces a sublinear time algorithm for approximating the spectral density of large graphs, achieving near-optimal accuracy with significantly reduced computational complexity.

Contribution

It presents a novel Chebyshev polynomial moment matching method that approximates spectral density efficiently, with theoretical guarantees and experimental validation.

Findings

Achieves $ ext{O}(n ext{poly}(1/ ext{epsilon}))$ runtime for spectral density approximation.

Proves that the method approximates spectral density within $ ext{epsilon}$ using $ ext{O}(1/ ext{epsilon})$ matrix-vector products.

Shows the method's stability and effectiveness with approximate matrix-vector products.

Abstract

We present a new sublinear time algorithm for approximating the spectral density (eigenvalue distribution) of an $n\times n$ normalized graph adjacency or Laplacian matrix. The algorithm recovers the spectrum up to $\epsilon$ accuracy in the Wasserstein-1 distance in $O(n\cdot \text{poly}(1/\epsilon))$ 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 $1/\epsilon$. 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 $\epsilon$ approximation to the spectral density using just $O({1}/{\epsilon})$ 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 \emph{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.