# Tight Error Bounds for the Sign-Constrained Stiefel Manifold

arXiv: https://arxiv.org/abs/2210.05164 · HTML: https://mlsanity.com/paper/2210.05164/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Oct 2022.

## Built on by (most cited first, 3 of 3)

- [Enhancing Distributional Robustness in Principal Component Analysis by Wasserstein Distances](https://mlsanity.com/paper/2503.02494/) (2025)
- [A Support-Set Algorithm for Optimization Problems with Nonnegative and Orthogonal Constraints](https://mlsanity.com/paper/2511.03443/) (2025)
- [Riemannian Difference-of-Convex Optimization for K-Means Clustering](https://mlsanity.com/paper/2609.34310/) (2026)

## Abstract (arXiv)

The sign-constrained Stiefel manifold in ℝn× 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 ℝn× 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 ℓ1 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.