论文标题
签名受限的stiefel歧管的严重错误界限
Tight Error Bounds for the Sign-Constrained Stiefel Manifold
论文作者
论文摘要
$ \ Mathbb {r}^{n \ times r} $中的符号约束的stiefel歧管是对矩阵的某些列的固定符号（非负或非阳性）的Stiefel歧管的一段。它包括非负态歧管作为特殊情况。我们提出了全局和本地误差界限，这些界限提供了一个易于计算的残差功能和显式系数的不等式，以绑定$ \ mathbb {r}^{n \ times r} $的矩阵的距离到签名受约束的stiefel歧管。此外，我们表明，除了在某些轻度条件下的乘法常数外，误差界不能改进，这解释了为什么当$ 1 <r <n $时，在界限中需要两个方界项，以及为什么当$ r = n $或$ r = 1 $的签名约束和Orthogogonations时，$ \ ell_1 $ norm可以在边界中使用。误差边界应用于得出精确的惩罚方法，以最大程度地限制Lipschitz连续功能，并具有正交性和符号约束。
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.
