# On a class of geodesically convex optimization problems solved via Euclidean MM methods

arXiv: https://arxiv.org/abs/2206.11426 · HTML: https://mlsanity.com/paper/2206.11426/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 52 references that are in the map.
- First posted on arXiv 22 Jun 2022.

## Builds on (most cited first, 19 of 19)

- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds](https://mlsanity.com/paper/1605.07147/) (2016)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Incremental Majorization-Minimization Optimization with Application to Large-Scale Machine Learning](https://mlsanity.com/paper/1402.4419/) (2014)
- [Geometric Mean Metric Learning](https://mlsanity.com/paper/1607.05002/) (2016)
- [Riemannian Optimization via Frank-Wolfe Methods](https://mlsanity.com/paper/1710.10770/) (2017)
- [Global Convergence of Non-Convex Gradient Descent for Computing Matrix Squareroot](https://mlsanity.com/paper/1507.05854/) (2015)
- [Fixed-point algorithms for learning determinantal point processes](https://mlsanity.com/paper/1508.00792/) (2015)
- [Regularized M-estimators of scatter matrix](https://mlsanity.com/paper/1405.2528/) (2014)
- [Much Faster Algorithms for Matrix Scaling](https://mlsanity.com/paper/1704.02315/) (2017)
- [Rigorous Guarantees for Tyler's M-estimator via quantum expansion](https://mlsanity.com/paper/2002.00071/) (2020)
- [Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods](https://mlsanity.com/paper/1704.02310/) (2017)
- [Operator Scaling via Geodesically Convex Optimization, Invariant Theory and Polynomial Identity Testing](https://mlsanity.com/paper/1804.01076/) (2018)
- [Algorithmic and optimization aspects of Brascamp-Lieb inequalities, via Operator Scaling](https://mlsanity.com/paper/1607.06711/) (2016)
- [On Geodesically Convex Formulations for the Brascamp-Lieb Constant](https://mlsanity.com/paper/1804.04051/) (2018)
- [On the matrix square root via geometric optimization](https://mlsanity.com/paper/1507.08366/) (2015)
- [Inference and Mixture Modeling with the Elliptical Gamma Distribution](https://mlsanity.com/paper/1410.4812/) (2014)

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

- [Changes from Classical Statistics to Modern Statistics and Data Science](https://mlsanity.com/paper/2211.03756/) (2022)

## Abstract (arXiv)

We study geodesically convex (g-convex) problems that can be written as a difference of Euclidean convex functions. This structure arises in several optimization problems in statistics and machine learning, e.g., for matrix scaling, M-estimators for covariances, and Brascamp-Lieb inequalities. Our work offers efficient algorithms that on the one hand exploit g-convexity to ensure global optimality along with guarantees on iteration complexity. On the other hand, the split structure permits us to develop Euclidean Majorization-Minorization algorithms that help us bypass the need to compute expensive Riemannian operations such as exponential maps and parallel transport. We illustrate our results by specializing them to a few concrete optimization problems that have been previously studied in the machine learning literature. Ultimately, we hope our work helps motivate the broader search for mixed Euclidean-Riemannian optimization algorithms