Exact Community Recovery (under Side Information): Optimality of Spectral Algorithms
<|reference_start|>Exact Community Recovery (under Side Information): Optimality of Spectral Algorithms: In this paper, we study the problem of exact community recovery in general, two-community block models considering both Bernoulli and Gaussian matrix models, capturing the Stochastic Block Model, submatrix localization, and $\mathbb{Z}_2$-synchronization as special cases. We also study the settings where $side$ $information$ about community assignment labels is available, modeled as passing the true labels through a noisy channel: either the binary erasure channel (where some community labels are known while others are erased) or the binary symmetric channel (where some labels are flipped). We provide a unified analysis of the effect of side information on the information-theoretic limits of exact recovery, generalizing prior works and extending to new settings. Additionally, we design a simple but optimal spectral algorithm that incorporates side information (when present) along with the eigenvectors of the matrix observation. Using the powerful tool of entrywise eigenvector analysis [Abbe, Fan, Wang, Zhong 2020], we show that our spectral algorithm can mimic the so called $genie$-$aided$ $estimators$, where the $i^{\mathrm{th}}$ genie-aided estimator optimally computes the estimate of the $i^{\mathrm{th}}$ label, when all remaining labels are revealed by a genie. This perspective provides a unified understanding of the optimality of spectral algorithms for various exact recovery problems in a recent line of work.<|reference_end|>
@article{gaudio2024exact,
    title={Exact Community Recovery (under Side Information): Optimality of
  Spectral Algorithms},
    author={Julia Gaudio, Nirmit Joshi},
    journal={arXiv preprint arXiv:2406.13075},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.13075},
    primaryClass={cs.SI cs.LG}
}