Tight Error Bounds for the Sign-Constrained Stiefel Manifold
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914801019518976 |
|---|---|
| author | Chen, Xiaojun He, Yifan Zhang, Zaikun |
| author_facet | Chen, Xiaojun He, Yifan Zhang, Zaikun |
| contents | The sign-constrained Stiefel manifold in $\mathbb{R}^{n\times r}$ is a segment of the Stiefel manifold with fixed signs (nonnegative or nonpositive) for some columns of the matrices. It includes the nonnegative Stiefel manifold as a special case. We present global and local error bounds that provide an inequality with easily computable residual functions and explicit coefficients to bound the distance from matrices in $\mathbb{R}^{n\times r}$ to the sign-constrained Stiefel manifold. Moreover, we show that the error bounds cannot be improved except for the multiplicative constants under some mild conditions, which explains why two square-root terms are necessary in the bounds when $1< r <n$ and why the $\ell_1$ norm can be used in the bounds when $r = n$ or $r = 1$ for the sign constraints and orthogonality, respectively. The error bounds are applied to derive exact penalty methods for minimizing a Lipschitz continuous function with orthogonality and sign constraints. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_05164 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Tight Error Bounds for the Sign-Constrained Stiefel Manifold Chen, Xiaojun He, Yifan Zhang, Zaikun Optimization and Control Numerical Analysis 65K05, 90C30, 90C26, 90C90 The sign-constrained Stiefel manifold in $\mathbb{R}^{n\times r}$ is a segment of the Stiefel manifold with fixed signs (nonnegative or nonpositive) for some columns of the matrices. It includes the nonnegative Stiefel manifold as a special case. We present global and local error bounds that provide an inequality with easily computable residual functions and explicit coefficients to bound the distance from matrices in $\mathbb{R}^{n\times r}$ to the sign-constrained Stiefel manifold. Moreover, we show that the error bounds cannot be improved except for the multiplicative constants under some mild conditions, which explains why two square-root terms are necessary in the bounds when $1< r <n$ and why the $\ell_1$ norm can be used in the bounds when $r = n$ or $r = 1$ for the sign constraints and orthogonality, respectively. The error bounds are applied to derive exact penalty methods for minimizing a Lipschitz continuous function with orthogonality and sign constraints. |
| title | Tight Error Bounds for the Sign-Constrained Stiefel Manifold |
| topic | Optimization and Control Numerical Analysis 65K05, 90C30, 90C26, 90C90 |
| url | https://arxiv.org/abs/2210.05164 |