# Convergence and complexity of block majorization-minimization for constrained block-Riemannian optimization

arXiv: https://arxiv.org/abs/2312.10330 · HTML: https://mlsanity.com/paper/2312.10330/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Dec 2023.

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

- [Manifold-based Algorithms for the Hadamard Decomposition](https://mlsanity.com/paper/2605.28980/) (2026)
- [Convergence and Complexity Guarantee for Inexact First-order Riemannian Optimization Algorithms](https://mlsanity.com/paper/2405.03073/) (2024)
- [A Spectral Framework for Tracking Communities in Evolving Networks](https://mlsanity.com/paper/2412.07378/) (2024)
- [Direction-Magnitude Decomposition for Low-Rank Matrix Optimization: Faster Convergence and Saddle-to-saddle Dynamics](https://mlsanity.com/paper/2606.31390/) (2026)

## Abstract (arXiv)

Block majorization-minimization (BMM) is a simple iterative algorithm for nonconvex optimization that sequentially minimizes a majorizing surrogate of the objective function in each block coordinate while the other block coordinates are held fixed. We consider a family of BMM algorithms for minimizing smooth nonconvex objectives, where each parameter block is constrained within a subset of a Riemannian manifold. We establish that this algorithm converges asymptotically to the set of stationary points, and attains an ε-stationary point within widetildeO(ε-2) iterations. In particular, the assumptions for our complexity results are completely Euclidean when the underlying manifold is a product of Euclidean or Stiefel manifolds, although our analysis makes explicit use of the Riemannian geometry. Our general analysis applies to a wide range of algorithms with Riemannian constraints: Riemannian MM, block projected gradient descent, optimistic likelihood estimation, geodesically constrained subspace tracking, robust PCA, and Riemannian CP-dictionary-learning. We experimentally validate that our algorithm converges faster than standard Euclidean algorithms applied to the Riemannian setting.