Faster Spectral Density Estimation and Sparsification in the Nuclear Norm
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| Format: | Preprint |
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2024
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| _version_ | 1866913386866933760 |
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| author | Jin, Yujia Karmarkar, Ishani Musco, Christopher Sidford, Aaron Singh, Apoorv Vikram |
| author_facet | Jin, Yujia Karmarkar, Ishani Musco, Christopher Sidford, Aaron Singh, Apoorv Vikram |
| contents | 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 $O(nε^{-2})$ queries to a degree and neighbor oracle and in $O(nε^{-3})$ time, estimates the spectrum up to $ε$ accuracy in the Wasserstein-1 metric. This improves on previous state-of-the-art methods, including an $O(nε^{-7})$ time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small $ε$, a $2^{O(ε^{-1})}$ time method from [Cohen-Steiner et al., KDD 2018]. To achieve this result, we introduce a new notion of graph sparsification, which we call nuclear sparsification. We provide an $O(nε^{-2})$-query and $O(nε^{-2})$-time algorithm for computing $O(nε^{-2})$-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 deterministic algorithm for spectral density estimation that scales linearly with $n$ (sublinear in the representation size of the graph). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_07521 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Faster Spectral Density Estimation and Sparsification in the Nuclear Norm Jin, Yujia Karmarkar, Ishani Musco, Christopher Sidford, Aaron Singh, Apoorv Vikram Data Structures and Algorithms Machine Learning 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 $O(nε^{-2})$ queries to a degree and neighbor oracle and in $O(nε^{-3})$ time, estimates the spectrum up to $ε$ accuracy in the Wasserstein-1 metric. This improves on previous state-of-the-art methods, including an $O(nε^{-7})$ time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small $ε$, a $2^{O(ε^{-1})}$ time method from [Cohen-Steiner et al., KDD 2018]. To achieve this result, we introduce a new notion of graph sparsification, which we call nuclear sparsification. We provide an $O(nε^{-2})$-query and $O(nε^{-2})$-time algorithm for computing $O(nε^{-2})$-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 deterministic algorithm for spectral density estimation that scales linearly with $n$ (sublinear in the representation size of the graph). |
| title | Faster Spectral Density Estimation and Sparsification in the Nuclear Norm |
| topic | Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2406.07521 |