Robust recovery for stochastic block models, simplified and generalized
CoRR(2024)
摘要
We study the problem of robust community recovery: efficiently
recovering communities in sparse stochastic block models in the presence of
adversarial corruptions. In the absence of adversarial corruptions, there are
efficient algorithms when the signal-to-noise ratio exceeds the
Kesten–Stigum (KS) threshold, widely believed to be the
computational threshold for this problem. The question we study is: does the
computational threshold for robust community recovery also lie at the KS
threshold? We answer this question affirmatively, providing an algorithm for
robust community recovery for arbitrary stochastic block models on any constant
number of communities, generalizing the work of Ding, d'Orsi, Nasser Steurer
on an efficient algorithm above the KS threshold in the case of 2-community
block models.
There are three main ingredients to our work:
(i) The Bethe Hessian of the graph is defined as H_G(t) ≜
(D_G-I)t^2 - A_Gt + I where D_G is the diagonal matrix of degrees and A_G
is the adjacency matrix. Empirical work suggested that the Bethe Hessian for
the stochastic block model has outlier eigenvectors corresponding to the
communities right above the Kesten-Stigum threshold. We formally confirm the
existence of outlier eigenvalues for the Bethe Hessian, by explicitly
constructing outlier eigenvectors from the community vectors.
(ii) We develop an algorithm for a variant of robust PCA on sparse matrices.
Specifically, an algorithm to partially recover top eigenspaces from
adversarially corrupted sparse matrices under mild delocalization constraints.
(iii) A rounding algorithm to turn vector assignments of vertices into a
community assignment, inspired by the algorithm of Charikar & Wirth
for 2XOR.
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