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

Melanie Weber; Suvrit Sra

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arXiv · 2206.11426

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

Melanie Weber·Suvrit Sra

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Abstract

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

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Melanie Weber, Suvrit Sra. 2022-06-22. On a class of geodesically convex optimization problems solved via Euclidean MM methods. https://arxiv.org/abs/2206.11426

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