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.

翻译：$\mathbb{R}^{n\times r}$ 中的符号约束Stiefel流形是Stiefel流形的一个子集，其矩阵某些列具有固定符号（非负或非正）。非负Stiefel流形是其特例。我们给出了全局和局部误差界，这些误差界通过易于计算的残差函数和显式系数建立了$\mathbb{R}^{n\times r}$中矩阵到符号约束Stiefel流形距离的不等式。此外，我们证明了在温和条件下，除了乘法常数外，该误差界无法改进，这解释了为何当$1< r <n$时界中需要两项平方根项，以及为何当$r=n$（符号约束）或$r=1$（正交性）时可在界中使用$\ell_1$范数。我们将这些误差界应用于推导求解具有正交性和符号约束的Lipschitz连续函数最小化问题的精确罚方法。