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Stochastic Gradient Descent on Riemannian Manifolds

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1 Author(s)
Bonnabel, S. ; Robot. Lab., Math. et Syst., Mines ParisTech, Paris, France

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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Automatic Control, IEEE Transactions on  (Volume:58 ,  Issue: 9 )