{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/benign-landscapes-of-low-dimensional","title":"Benign landscapes of low-dimensional relaxations for orthogonal synchronization on general graphs","arxiv_id":"2307.02941","date":"2023-07-06","proceeding":null,"authors":["Andrew D. McRae","Nicolas Boumal"],"abstract":"Orthogonal group synchronization is the problem of estimating $n$ elements $Z_1, \\ldots, Z_n$ from the $r \\times r$ orthogonal group given some relative measurements $R_{ij} \\approx Z_i^{}Z_j^{-1}$. The least-squares formulation is nonconvex. To avoid its local minima, a Shor-type convex relaxation squares the dimension of the optimization problem from $O(n)$ to $O(n^2)$. Alternatively, Burer--Monteiro-type nonconvex relaxations have generic landscape guarantees at dimension $O(n^{3/2})$. For smaller relaxations, the problem structure matters. It has been observed in the robotics literature that, for SLAM problems, it seems sufficient to increase the dimension by a small constant multiple over the original. We partially explain this. This also has implications for Kuramoto oscillators. Specifically, we minimize the least-squares cost function in terms of estimators $Y_1, \\ldots, Y_n$. For $p \\geq r$, each $Y_i$ is relaxed to the Stiefel manifold $\\mathrm{St}(r, p)$ of $r \\times p$ matrices with orthonormal rows. The available measurements implicitly define a (connected) graph $G$ on $n$ vertices. In the noiseless case, we show that, for all connected graphs $G$, second-order critical points are globally optimal as soon as $p \\geq r+2$. (This implies that Kuramoto oscillators on $\\mathrm{St}(r, p)$ synchronize for all $p \\geq r + 2$.) This result is the best possible for general graphs; the previous best known result requires $2p \\geq 3(r + 1)$. For $p > r + 2$, our result is robust to modest amounts of noise (depending on $p$ and $G$). Our proof uses a novel randomized choice of tangent direction to prove (near-)optimality of second-order critical points. Finally, we partially extend our noiseless landscape results to the complex case (unitary group); we show that there are no spurious local minima when $2p \\geq 3r$.","url_abs":"https://arxiv.org/abs/2307.02941v2","url_pdf":"https://arxiv.org/pdf/2307.02941v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"benign-landscapes-of-low-dimensional","repo_url":"https://github.com/admcrae/sync_relaxed","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2307.02941","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}