# A Manifold Proximal Linear Method for Sparse Spectral Clustering with Application to Single-Cell RNA Sequencing Data Analysis

arXiv: https://arxiv.org/abs/2007.09524 · HTML: https://mlsanity.com/paper/2007.09524/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jul 2020.

## Built on by (most cited first, 7 of 7)

- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [A Riemannian ADMM](https://mlsanity.com/paper/2211.02163/) (2022)
- [A Riemannian smoothing steepest descent method for non-Lipschitz optimization on submanifolds](https://mlsanity.com/paper/2104.04199/) (2021)
- [A New Inexact Proximal Linear Algorithm with Adaptive Stopping Criteria for Robust Phase Retrieval](https://mlsanity.com/paper/2304.12522/) (2023)
- [Do algorithms and barriers for sparse principal component analysis extend to other structured settings?](https://mlsanity.com/paper/2307.13535/) (2023)
- [Provably Uncertainty-Guided Universal Domain Adaptation](https://mlsanity.com/paper/2209.09616/) (2022)
- [Enhancing Distributional Robustness in Principal Component Analysis by Wasserstein Distances](https://mlsanity.com/paper/2503.02494/) (2025)

## Abstract (arXiv)

Spectral clustering is one of the fundamental unsupervised learning methods widely used in data analysis. Sparse spectral clustering (SSC) imposes sparsity to the spectral clustering and it improves the interpretability of the model. This paper considers a widely adopted model for SSC, which can be formulated as an optimization problem over the Stiefel manifold with nonsmooth and nonconvex objective. Such an optimization problem is very challenging to solve. Existing methods usually solve its convex relaxation or need to smooth its nonsmooth part using certain smoothing techniques. In this paper, we propose a manifold proximal linear method (ManPL) that solves the original SSC formulation. We also extend the algorithm to solve the multiple-kernel SSC problems, for which an alternating ManPL algorithm is proposed. Convergence and iteration complexity results of the proposed methods are established. We demonstrate the advantage of our proposed methods over existing methods via the single-cell RNA sequencing data analysis.