# A Riemannian ADMM

arXiv: https://arxiv.org/abs/2211.02163 · HTML: https://mlsanity.com/paper/2211.02163/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 61 references that are in the map.
- First posted on arXiv 3 Nov 2022.

## Builds on (most cited first, 26 of 26)

- [Global Convergence of ADMM in Nonconvex Nonsmooth Optimization](https://mlsanity.com/paper/1511.06324/) (2015)
- [A Geometric Analysis of Phase Retrieval](https://mlsanity.com/paper/1602.06664/) (2016)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Complete Dictionary Recovery over the Sphere I: Overview and the Geometric Picture](https://mlsanity.com/paper/1511.03607/) (2015)
- [An Overview of Robust Subspace Recovery](https://mlsanity.com/paper/1803.01013/) (2018)
- [Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices](https://mlsanity.com/paper/1507.02772/) (2015)
- [Complete Dictionary Recovery over the Sphere II: Recovery by Riemannian Trust-region Method](https://mlsanity.com/paper/1511.04777/) (2015)
- [Douglas-Rachford splitting and ADMM for nonconvex optimization: tight convergence results](https://mlsanity.com/paper/1709.05747/) (2017)
- [Fast, Robust and Non-convex Subspace Recovery](https://mlsanity.com/paper/1406.6145/) (2014)
- [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)
- [A Well-Tempered Landscape for Non-convex Robust Subspace Recovery](https://mlsanity.com/paper/1706.03896/) (2017)
- [Structured Nonconvex and Nonsmooth Optimization: Algorithms and Iteration Complexity Analysis](https://mlsanity.com/paper/1605.02408/) (2016)
- [MADMM: a generic algorithm for non-smooth optimization on manifolds](https://mlsanity.com/paper/1505.07676/) (2015)
- [Moreau Envelope Augmented Lagrangian Method for Nonconvex Optimization with Linear Constraints](https://mlsanity.com/paper/2101.08519/) (2021)
- [Scaling Law for Recovering the Sparsest Element in a Subspace](https://mlsanity.com/paper/1310.1654/) (2013)
- [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)
- [Primal-Dual Optimization Algorithms over Riemannian Manifolds: an Iteration Complexity Analysis](https://mlsanity.com/paper/1710.02236/) (2017)
- [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)
- [On the Iteration Complexity of Smoothed Proximal ALM for Nonconvex Optimization Problem with Convex Constraints](https://mlsanity.com/paper/2207.06304/) (2022)
- [Alternating Direction Method of Multipliers for A Class of Nonconvex and Nonsmooth Problems with Applications to Background/Foreground Extraction](https://mlsanity.com/paper/1506.07029/) (2015)
- [Stochastic and Private Nonconvex Outlier-Robust PCA](https://mlsanity.com/paper/2203.09276/) (2022)
- [A Regularized Semi-Smooth Newton Method With Projection Steps for Composite Convex Programs](https://mlsanity.com/paper/1603.07870/) (2016)
- [An Extension of Fast Iterative Shrinkage-thresholding to Riemannian Optimization for Sparse Principal Component Analysis](https://mlsanity.com/paper/1909.05485/) (2019)
- [Nonconvex Sparse Spectral Clustering by Alternating Direction Method of Multipliers and Its Convergence Analysis](https://mlsanity.com/paper/1712.02979/) (2017)

## Built on by (most cited first, 3 of 3)

- [Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds](https://mlsanity.com/paper/2510.21468/) (2025)
- [ADMM for Structured Fractional Minimization](https://mlsanity.com/paper/2411.07496/) (2024)
- [Balancing Latency and Model Accuracy for Fluid Antenna-Assisted LM-Embedded MIMO Network](https://mlsanity.com/paper/2508.01689/) (2025)

## Abstract (arXiv)

We consider a class of Riemannian optimization problems where the objective is the sum of a smooth function and a nonsmooth function, considered in the ambient space. This class of problems finds important applications in machine learning and statistics such as the sparse principal component analysis, sparse spectral clustering, and orthogonal dictionary learning. We propose a Riemannian alternating direction method of multipliers (ADMM) to solve this class of problems. Our algorithm adopts easily computable steps in each iteration. The iteration complexity of the proposed algorithm for obtaining an ε-stationary point is analyzed under mild assumptions. Existing ADMM for solving nonconvex problems either does not allow nonconvex constraint set, or does not allow nonsmooth objective function. Our algorithm is the first ADMM type algorithm that minimizes a nonsmooth objective over manifold -- a particular nonconvex set. Numerical experiments are conducted to demonstrate the advantage of the proposed method.