# A Decentralized Proximal Gradient Tracking Algorithm for Composite Optimization on Riemannian Manifolds

arXiv: https://arxiv.org/abs/2401.11573 · HTML: https://mlsanity.com/paper/2401.11573/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jan 2024.

## Built on by (most cited first, 4 of 4)

- [Convergence of Decentralized Stochastic Subgradient-based Methods for Nonsmooth Nonconvex functions](https://mlsanity.com/paper/2403.11565/) (2024)
- [A Double Tracking Method for Optimization with Decentralized Generalized Orthogonality Constraints](https://mlsanity.com/paper/2409.04998/) (2024)
- [Enhancing Distributional Robustness in Principal Component Analysis by Wasserstein Distances](https://mlsanity.com/paper/2503.02494/) (2025)
- [A Support-Set Algorithm for Optimization Problems with Nonnegative and Orthogonal Constraints](https://mlsanity.com/paper/2511.03443/) (2025)

## Abstract (arXiv)

This paper focuses on minimizing a smooth function combined with a nonsmooth regularization term on a compact Riemannian submanifold embedded in the Euclidean space under a decentralized setting. Typically, there are two types of approaches at present for tackling such composite optimization problems. The first, subgradient-based approaches, rely on subgradient information of the objective function to update variables, achieving an iteration complexity of O(ε-4log2(ε-2)). The second, smoothing approaches, involve constructing a smooth approximation of the nonsmooth regularization term, resulting in an iteration complexity of O(ε-4). This paper proposes a proximal gradient type algorithm that fully exploits the composite structure. The global convergence to a stationary point is established with a significantly improved iteration complexity of O(ε-2). To validate the effectiveness and efficiency of our proposed method, we present numerical results in real-world applications, showcasing its superior performance.