# A Riemannian smoothing steepest descent method for non-Lipschitz optimization on submanifolds

arXiv: https://arxiv.org/abs/2104.04199 · HTML: https://mlsanity.com/paper/2104.04199/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 54 references that are in the map.
- First posted on arXiv 9 Apr 2021.

## Builds on (most cited first, 13 of 13)

- [Complete Dictionary Recovery over the Sphere I: Overview and the Geometric Picture](https://mlsanity.com/paper/1511.03607/) (2015)
- [A note on the sample complexity of the Er-SpUD algorithm by Spielman, Wang and Wright for exact recovery of sparsely used dictionaries](https://mlsanity.com/paper/1601.02049/) (2016)
- [Dual Principal Component Pursuit](https://mlsanity.com/paper/1510.04390/) (2015)
- [Finding a sparse vector in a subspace: Linear sparsity using alternating directions](https://mlsanity.com/paper/1412.4659/) (2014)
- [Bilinear Factor Matrix Norm Minimization for Robust PCA: Algorithms and Applications](https://mlsanity.com/paper/1810.05186/) (2018)
- [Finding the Sparsest Vectors in a Subspace: Theory, Algorithms, and Applications](https://mlsanity.com/paper/2001.06970/) (2020)
- [MADMM: a generic algorithm for non-smooth optimization on manifolds](https://mlsanity.com/paper/1505.07676/) (2015)
- [A Manifold Proximal Linear Method for Sparse Spectral Clustering with Application to Single-Cell RNA Sequencing Data Analysis](https://mlsanity.com/paper/2007.09524/) (2020)
- [Manifold Proximal Point Algorithms for Dual Principal Component Pursuit and Orthogonal Dictionary Learning](https://mlsanity.com/paper/2005.02356/) (2020)
- [A Semismooth Newton based Augmented Lagrangian Method for Nonsmooth Optimization on Matrix Manifolds](https://mlsanity.com/paper/2103.02855/) (2021)
- [Riemannian Stochastic Proximal Gradient Methods for Nonsmooth Optimization over the Stiefel Manifold](https://mlsanity.com/paper/2005.01209/) (2020)
- [Joint Power and Admission Control: Non-Convex Lq Approximation and An Effective Polynomial Time Deflation Approach](https://mlsanity.com/paper/1311.3045/) (2013)
- [Smoothing SQP methods for solving degenerate nonsmooth constrained optimization problems with applications to bilevel programs](https://mlsanity.com/paper/1403.1636/) (2014)

## Built on by (most cited first, 1 of 1)

- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)

## Abstract (arXiv)

In this paper, we propose a Riemannian smoothing steepest descent method to minimize a nonconvex and non-Lipschitz function on submanifolds. The generalized subdifferentials on Riemannian manifold and the Riemannian gradient sub-consistency are defined and discussed. We prove that any accumulation point of the sequence generated by the Riemannian smoothing steepest descent method is a stationary point associated with the smoothing function employed in the method, which is necessary for the local optimality of the original non-Lipschitz problem. Under the Riemannian gradient sub-consistency condition, we also prove that any accumulation point is a Riemannian limiting stationary point of the original non-Lipschitz problem. Numerical experiments are conducted to demonstrate the efficiency of the proposed method.