{
  "id": "2303.17001",
  "version": "v1",
  "published": "2023-03-29T20:07:07.000Z",
  "updated": "2023-03-29T20:07:07.000Z",
  "title": "The G-invariant graph Laplacian",
  "authors": [
    "Eitan Rosen",
    "Yoel Shkolnisky"
  ],
  "categories": [
    "cs.LG",
    "cs.SI"
  ],
  "abstract": "Graph Laplacian based algorithms for data lying on a manifold have been proven effective for tasks such as dimensionality reduction, clustering, and denoising. In this work, we consider data sets whose data point not only lie on a manifold, but are also closed under the action of a continuous group. An example of such data set is volumes that line on a low dimensional manifold, where each volume may be rotated in three-dimensional space. We introduce the G-invariant graph Laplacian that generalizes the graph Laplacian by accounting for the action of the group on the data set. We show that like the standard graph Laplacian, the G-invariant graph Laplacian converges to the Laplace-Beltrami operator on the data manifold, but with a significantly improved convergence rate. Furthermore, we show that the eigenfunctions of the G-invariant graph Laplacian admit the form of tensor products between the group elements and eigenvectors of certain matrices, which can be computed efficiently using FFT-type algorithms. We demonstrate our construction and its advantages on the problem of filtering data on a noisy manifold closed under the action of the special unitary group SU(2).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-29T20:07:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "data set",
      "special unitary group su",
      "g-invariant graph laplacian admit",
      "g-invariant graph laplacian converges",
      "low dimensional manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}