Transferability of Error Bounds and the Kurdyka-Łojasiewicz Property under C1 Partial Smoothness
Abstract
This work studies how a generalized error bound (GEB) property in the ambient Euclidean space transfers to the active manifold under C1 partial smoothness. The GEB generalizes the standard error bound, which upper bounds the distance to a subset of critical points using first-order stationarity residuals, by allowing the distance to be composed with a gauge function. In the ambient space, these residuals are measured by the subdifferential or the slope of the original function, while on the manifold they are measured by the Riemannian gradient or the slope of the restricted function. We develop complementary geometric and metric analyses to show that the ambient and manifold GEBs are equivalent with the same gauge function. The geometric argument shows that the projection of the subdifferential onto the tangent space coincides with the Riemannian gradient, while the metric approach combines slope estimates, identifiability, and a new uniform linear-growth result for C1 manifolds. We further extend our analysis to establish the analogous equivalence for the Kurdyka--Łojasiewicz (KL) property, with the same desingularizer, and separately characterize their connections with the GEB. Finally, for regularized optimization, we relate subdifferential and proximal error bounds, thereby placing the previously established equivalence between proximal and manifold error bounds for ℓ1-regularization within a broader framework.
Related
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BibTeX
@misc{wu2026transferability,
title = {{Transferability of Error Bounds and the Kurdyka-Łojasiewicz Property under $C^1$ Partial Smoothness}},
author = {Hanju Wu and Ching-Pei Lee and Yue Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.02754},
eprint = {2605.02754},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.02754},
}
RIS
TY - UNPB
TI - Transferability of Error Bounds and the Kurdyka-Łojasiewicz Property under C^1 Partial Smoothness
AU - Hanju Wu
AU - Ching-Pei Lee
AU - Yue Xie
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2605.02754
UR - https://arxiv.org/abs/2605.02754
AB - This work studies how a generalized error bound (GEB) property in the ambient Euclidean space transfers to the active manifold under C^1 partial smoothness. The GEB generalizes the standard error bound, which upper bounds the distance to a subset of critical points using first-order stationarity residuals, by allowing the distance to be composed with a gauge function. In the ambient space, these residuals are measured by the subdifferential or the slope of the original function, while on the manifold they are measured by the Riemannian gradient or the slope of the restricted function. We develop complementary geometric and metric analyses to show that the ambient and manifold GEBs are equivalent with the same gauge function. The geometric argument shows that the projection of the subdifferential onto the tangent space coincides with the Riemannian gradient, while the metric approach combines slope estimates, identifiability, and a new uniform linear-growth result for C^1 manifolds. We further extend our analysis to establish the analogous equivalence for the Kurdyka--Łojasiewicz (KL) property, with the same desingularizer, and separately characterize their connections with the GEB. Finally, for regularized optimization, we relate subdifferential and proximal error bounds, thereby placing the previously established equivalence between proximal and manifold error bounds for ℓ_1-regularization within a broader framework.
ER -