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We consider the problem of estimating the spectral density of the normalized adjacency matrix of an $$n$$-node undirected graph. We provide a randomized algorithm that, with $$O(n\epsilon^{-2})$$ queries to a degree and neighbor oracle and in $$O(n\epsilon^{-3})$$ time, estimates the spectrum up to $$\epsilon$$ accuracy in the Wasserstein-1 metric. This improves on previous state-of-the-art methods, including an $$O(n\epsilon^{-7})$$ time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small $$\epsilon$$, a $$2^{O(\epsilon^{-1})}$$ time method from [Cohen-Steiner et al., KDD 2018]. To achieve this result, we introduce a new notion of graph sparsification, which we call \emph{nuclear sparsification}. We provide an $$O(n\epsilon^{-2})$$-query and $$O(n\epsilon^{-2})$$-time algorithm for computing $$O(n\epsilon^{-2})$$-sparse nuclear sparsifiers. We show that this bound is optimal in both its sparsity and query complexity, and we separate our results from the related notion of additive spectral sparsification. Of independent interest, we show that our sparsification method also yields the first \emph{deterministic} algorithm for spectral density estimation that scales linearly with $$n$$ (sublinear in the representation size of the graph).
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Moments, Random Walks, and Limits for Spectrum Approximation
We study lower bounds for the problem of approximating a one dimensional distribution given (noisy) measurements of its moments. We show that there are distributions on $[-1,1]$ that cannot be approximated to accuracy $$\epsilon$$ in Wasserstein-1 distance even if we know \emph{all} of their moments to multiplicative accuracy $$(1\pm2^{-\Omega(1/\epsilon)})$$; this result matches an upper bound of Kong and Valiant [Annals of Statistics, 2017]. To obtain our result, we provide a hard instance involving distributions induced by the eigenvalue spectra of carefully constructed graph adjacency matrices. Efficiently approximating such spectra in Wasserstein-1 distance is a well-studied algorithmic problem, and a recent result of Cohen-Steiner et al. [KDD 2018] gives a method based on accurately approximating spectral moments using $$2^{O(1/\epsilon)}$$ random walks initiated at uniformly random nodes in the graph.As a strengthening of our main result, we show that improving the dependence on $$1/\epsilon$$ in this result would require a new algorithmic approach. Specifically, no algorithm can compute an $$\epsilon$$-accurate approximation to the spectrum of a normalized graph adjacency matrix with constant probability, even when given the transcript of $$2^{\Omega(1/\epsilon)}$$ random walks of length $$2^{\Omega(1/\epsilon)}$$ started at random nodes.
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- Award ID(s):
- 2045590
- PAR ID:
- 10495903
- Publisher / Repository:
- Proceedings of Machine Learning Research
- Date Published:
- Journal Name:
- Proceedings of Thirty Sixth Conference on Learning Theory
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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