{
  "id": "1407.5178",
  "version": "v1",
  "published": "2014-07-19T12:19:39.000Z",
  "updated": "2014-07-19T12:19:39.000Z",
  "title": "Properties and Applications of a Restricted HR Gradient Operator",
  "authors": [
    "Mengdi Jiang",
    "Yi Li",
    "Wei Liu"
  ],
  "comment": "This paper is about a quaternion-valued gradient operator and its properties and applications",
  "categories": [
    "math.OC"
  ],
  "abstract": "For quaternionic signal processing algorithms, the gradients of a quaternion-valued function are required for gradient-based methods. Given the non-commutativity of quaternion algebra, the definition of the gradients is non-trivial. The HR gradient operator provides a viable framework and has found a number of applications. However, the applications so far have been mainly limited to real-valued quaternion functions and linear quaternion-valued functions. To generalize the operator to nonlinear quaternion functions, we define a restricted version of the HR operator. The restricted HR gradient operator comes in two versions, the left and the right ones. We then present a detailed analysis of the properties of the operators, including several different product rules and chain rules. Using the new rules, we derive explicit expressions for the derivatives of a class of regular nonlinear quaternion-valued functions, and prove that the restricted HR gradients are consistent with the gradients in real domain.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-19T12:19:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "applications",
      "properties",
      "restricted hr gradient operator comes",
      "quaternionic signal processing algorithms",
      "regular nonlinear quaternion-valued functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.5178J"
    }
  }
}