Nonconvex landscapes for $\mathbf{Z}_2$ synchronization and graph clustering are benign near exact recovery thresholds

[2407.13407] Nonconvex landscapes for $\mathbf{Z}_2$ synchronization and graph clustering are benign near exact recovery thresholds Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Optimization and Control arXiv:2407.13407 (math) [Submitted on 18 Jul 2024] Title: Nonconvex landscapes for $\mathbf{Z}_2$ synchronization and graph clustering are benign near exact recovery thresholds Authors: Andrew D. McRae , Pedro Abdalla , Afonso S. Bandeira , Nicolas Boumal View a PDF of the paper titled Nonconvex landscapes for $\mathbf{Z}_2$ synchronization and graph clustering are benign near exact recovery thresholds, by Andrew D. McRae and Pedro Abdalla and Afonso S. Bandeira and Nicolas Boumal View PDF HTML (experimental) Abstract: We study the optimization landscape of a smooth nonconvex program arising from synchronization over the two-element group $\mathbf{Z}_2$, that is, recovering $z_1, \dots, z_n \in \{\pm 1\}$ from (noisy) relative measurements $R_{ij} \approx z_i z_j$. Starting from a max-cut--like combinatorial problem, for integer parameter $r \geq 2$, the nonconvex problem we study can be viewed both as a rank-$r$ Burer--Monteiro factorization of the standard max-cut semidefinite relaxation and as a relaxation of $\{ \pm 1 \}$ to the unit sphere in $\mathbf{R}^r$. First, we present deterministic, non-asymptotic conditions on the measurement graph and noise under which every second-order critical point of the nonconvex problem yields exact recovery of the ground truth. Then, via probabilistic analysis, we obtain asymptotic guarantees for three benchmark problems: (1) synchronization with a complete graph and Gaussian noise, (2) synchronization with an Erdős--Rényi random graph and Bernoulli noise, and (3) graph clustering under the binary symmetric stochastic block model. In each case, we have, asymptotically as the problem size goes to infinity, a benign nonconvex landscape near a previously-established optimal threshold for exact recovery; we can approach this threshold to arbitrary precision with large enough (but finite) rank parameter $r$. In addition, our results are robust to monotone adversaries. Subjects: Optimization and Control (math.OC) ; Statistics Theory (math.ST) Cite as: arXiv:2407.13407 [math.OC] (or arXiv:2407.13407v1 [math.OC] for this version) https://doi.org/10.48550/arXiv.2407.13407 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1093/imaiai/iaaf012 Focus to learn more DOI(s) linking to related resources Submission history From: Andrew McRae [ view email ] [v1] Thu, 18 Jul 2024 11:21:58 UTC (27 KB) Full-text links: Access Paper: View a PDF of the paper titled Nonconvex landscapes for $\mathbf{Z}_2$ synchronization and graph clustering are benign near exact recovery thresholds, by Andrew D. McRae and Pedro Abdalla and Afonso S. Bandeira and Nicolas Boumal View PDF HTML (experimental) TeX Source view license Current browse context: math.OC < prev | next > new | recent | 2024-07 Change to browse by: math math.ST stat stat.TH References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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