Global convergence of the subgradient method for robust signal recovery
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908778483417088 |
|---|---|
| author | Cai, Zesheng Lai, Lexiao Li, Tiansheng |
| author_facet | Cai, Zesheng Lai, Lexiao Li, Tiansheng |
| contents | We study the subgradient method for factorized robust signal recovery problems, including robust PCA, robust phase retrieval, and robust matrix sensing. The resulting objectives are nonsmooth and nonconvex, and can have unbounded sublevel sets, so standard analyses based on descent and coercivity do not apply. For locally Lipschitz semialgebraic objectives, we develop a convergence framework that replaces these requirements with a boundedness condition on continuous-time subgradient trajectories. Under this condition and sufficiently small step sizes of order $1/k$, we show that iterates of the subgradient method remain bounded and the full sequence converges to a critical point. We then verify the required boundedness property for the three robust objectives by adapting existing trajectory analyses, assuming a mild nondegeneracy condition in the matrix sensing case. Finally, for rank-one symmetric robust PCA, we prove that for almost every initialization, the method cannot converge to spurious critical points; consequently, under the same step-size regime, it converges to a global minimum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10062 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global convergence of the subgradient method for robust signal recovery Cai, Zesheng Lai, Lexiao Li, Tiansheng Optimization and Control 90C26, 49J52, 94A12 We study the subgradient method for factorized robust signal recovery problems, including robust PCA, robust phase retrieval, and robust matrix sensing. The resulting objectives are nonsmooth and nonconvex, and can have unbounded sublevel sets, so standard analyses based on descent and coercivity do not apply. For locally Lipschitz semialgebraic objectives, we develop a convergence framework that replaces these requirements with a boundedness condition on continuous-time subgradient trajectories. Under this condition and sufficiently small step sizes of order $1/k$, we show that iterates of the subgradient method remain bounded and the full sequence converges to a critical point. We then verify the required boundedness property for the three robust objectives by adapting existing trajectory analyses, assuming a mild nondegeneracy condition in the matrix sensing case. Finally, for rank-one symmetric robust PCA, we prove that for almost every initialization, the method cannot converge to spurious critical points; consequently, under the same step-size regime, it converges to a global minimum. |
| title | Global convergence of the subgradient method for robust signal recovery |
| topic | Optimization and Control 90C26, 49J52, 94A12 |
| url | https://arxiv.org/abs/2601.10062 |