{
  "id": "1111.5280",
  "version": "v4",
  "published": "2011-11-22T18:41:12.000Z",
  "updated": "2013-11-19T11:56:10.000Z",
  "title": "Stochastic gradient descent on Riemannian manifolds",
  "authors": [
    "Silvere Bonnabel"
  ],
  "comment": "A slightly shorter version has been published in IEEE Transactions Automatic Control",
  "journal": "IEEE Transactions on Automatic Control, Vol 58 (9), pages 2217 - 2229, Sept 2013",
  "categories": [
    "math.OC",
    "cs.LG",
    "stat.ML"
  ],
  "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.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-11-19T11:56:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riemannian manifold",
      "cost function",
      "procedure extending stochastic gradient descent",
      "gradient descent algorithm converges",
      "extending stochastic gradient descent algorithms"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.5280B"
    }
  }
}