Tight Error Bounds for the Sign-Constrained Stiefel Manifold

2022/10/11 by Xiaojun Chen, Yifan He, Chen, Xiaojun +3 · 2 citations
Mathematics · Engineering · Computer Science · #Advanced Optimization Algorithms Research #Advanced Numerical Analysis Techniques #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.2210.05164

Abstract

The sign-constrained Stiefel manifold in ℝ^{n× 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 ℝ^{n× 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

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• A Support-Set Algorithm for Optimization Problems with Nonnegative and Orthogonal Constraints
• An Inexact Proximal Framework for Nonsmooth Riemannian Difference-of-Convex Optimization

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