{
  "id": "2407.02082",
  "version": "v1",
  "published": "2024-07-02T09:19:20.000Z",
  "updated": "2024-07-02T09:19:20.000Z",
  "title": "Conditioning the complexity of random landscapes on marginal optima",
  "authors": [
    "Jaron Kent-Dobias"
  ],
  "categories": [
    "cond-mat.dis-nn",
    "cond-mat.stat-mech"
  ],
  "abstract": "Marginal optima are minima or maxima of a function with many nearly flat directions. In settings with many competing optima, marginal ones tend to attract algorithms and physical dynamics. Often, the important family of marginal attractors are a vanishing minority compared with nonmarginal optima and other unstable stationary points. We introduce a generic technique for conditioning the statistics of stationary points in random landscapes on their marginality, and apply it in three isotropic settings with qualitatively different structure: in the spherical spin-glasses, where the energy is Gaussian and its Hessian is GOE; in multispherical spin glasses, which are Gaussian but non-GOE; and in sums of squared spherical random functions, which are non-Gaussian. In these problems we are able to fully characterize the distribution of marginal optima in the landscape, including when they are in the minority.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-02T09:19:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random landscapes",
      "complexity",
      "conditioning",
      "attract algorithms",
      "marginal attractors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}