{
  "id": "2310.13821",
  "version": "v1",
  "published": "2023-10-20T21:18:04.000Z",
  "updated": "2023-10-20T21:18:04.000Z",
  "title": "Geometric Learning with Positively Decomposable Kernels",
  "authors": [
    "Nathael Da Costa",
    "Cyrus Mostajeran",
    "Juan-Pablo Ortega",
    "Salem Said"
  ],
  "categories": [
    "cs.LG",
    "math.DG",
    "stat.ML"
  ],
  "abstract": "Kernel methods are powerful tools in machine learning. Classical kernel methods are based on positive-definite kernels, which map data spaces into reproducing kernel Hilbert spaces (RKHS). For non-Euclidean data spaces, positive-definite kernels are difficult to come by. In this case, we propose the use of reproducing kernel Krein space (RKKS) based methods, which require only kernels that admit a positive decomposition. We show that one does not need to access this decomposition in order to learn in RKKS. We then investigate the conditions under which a kernel is positively decomposable. We show that invariant kernels admit a positive decomposition on homogeneous spaces under tractable regularity assumptions. This makes them much easier to construct than positive-definite kernels, providing a route for learning with kernels for non-Euclidean data. By the same token, this provides theoretical foundations for RKKS-based methods in general.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-20T21:18:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positively decomposable kernels",
      "positive-definite kernels",
      "geometric learning",
      "kernel methods",
      "kernel hilbert spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}