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arXiv · 1111.5280

Stochastic gradient descent on Riemannian manifolds

Silvere Bonnabel
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Abstract

Stochastic gradient descent is a simple approach to find the local minima of a cost function whose evaluations are corrupted by noise. In this paper, we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a Riemannian manifold. We prove that, as in the Euclidian case, the gradient descent algorithm converges to a critical point of the cost function. The algorithm has numerous potential applications, and is illustrated here by four examples. In particular a novel gossip algorithm on the set of covariance matrices is derived and tested numerically.

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Silvere Bonnabel. 2013-11-19. Stochastic gradient descent on Riemannian manifolds. https://doi.org/10.1109/tac.2013.2254619

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