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

• URL: http://arxiv.org/abs/2206.11426v1
• Date: Wed, 22 Jun 2022 23:57:40 GMT
• Title: On a class of geodesically convex optimization problems solved via Euclidean MM methods
• Authors: Suvrit Sra and Melanie Weber
• Abstract summary: We show how a difference of Euclidean convexization functions can be written as a difference of different types of problems in statistics and machine learning. Ultimately, we helps the broader broader the broader the broader the broader the work.
• Score: 50.428784381385164
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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