# Riemannian Smoothing Gradient Type Algorithms]{Riemannian Smoothing Gradient Type Algorithms for Nonsmooth Optimization Problem on Compact Riemannian Submanifold Embedded in Euclidean Space

arXiv: https://arxiv.org/abs/2212.03526 · HTML: https://mlsanity.com/paper/2212.03526/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Dec 2022.

## Built on by (most cited first, 2 of 2)

- [Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds](https://mlsanity.com/paper/2510.21468/) (2025)
- [Primal-Dual Methods for Nonsmooth Nonconvex Optimization with Orthogonality Constraints](https://mlsanity.com/paper/2604.04130/) (2026)

## Abstract (arXiv)

In this paper, we introduce the notion of generalized ε-stationarity for a class of nonconvex and nonsmooth composite minimization problems on compact Riemannian submanifold embedded in Euclidean space. To find a generalized ε-stationarity point, we develop a family of Riemannian gradient-type methods based on the Moreau envelope technique with a decreasing sequence of smoothing parameters, namely Riemannian smoothing gradient and Riemannian smoothing stochastic gradient methods. We prove that the Riemannian smoothing gradient method has the iteration complexity of O(ε-3) for driving a generalized ε-stationary point. To our knowledge, this is the best-known iteration complexity result for the nonconvex and nonsmooth composite problem on manifolds. For the Riemannian smoothing stochastic gradient method, one can achieve the iteration complexity of O(ε-5) for driving a generalized ε-stationary point. Numerical experiments are conducted to validate the superiority of our algorithms.