On a class of geodesically convex optimization problems solved via Euclidean MM methods
This work provides incremental improvements for researchers in optimization, statistics, and machine learning by offering more efficient algorithms for specific geodesically convex problems.
The paper tackles geodesically convex optimization problems that can be expressed as a difference of Euclidean convex functions, common in areas like matrix scaling and M-estimators, by developing efficient Euclidean Majorization-Minorization algorithms that avoid costly Riemannian operations while ensuring global optimality and iteration complexity guarantees.
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