{
  "id": "2407.03608",
  "version": "v1",
  "published": "2024-07-04T03:39:55.000Z",
  "updated": "2024-07-04T03:39:55.000Z",
  "title": "Gaussian process regression with log-linear scaling for common non-stationary kernels",
  "authors": [
    "P. Michael Kielstra",
    "Michael Lindsey"
  ],
  "categories": [
    "math.NA",
    "cs.NA",
    "stat.CO"
  ],
  "abstract": "We introduce a fast algorithm for Gaussian process regression in low dimensions, applicable to a widely-used family of non-stationary kernels. The non-stationarity of these kernels is induced by arbitrary spatially-varying vertical and horizontal scales. In particular, any stationary kernel can be accommodated as a special case, and we focus especially on the generalization of the standard Mat\\'ern kernel. Our subroutine for kernel matrix-vector multiplications scales almost optimally as $O(N\\log N)$, where $N$ is the number of regression points. Like the recently developed equispaced Fourier Gaussian process (EFGP) methodology, which is applicable only to stationary kernels, our approach exploits non-uniform fast Fourier transforms (NUFFTs). We offer a complete analysis controlling the approximation error of our method, and we validate the method's practical performance with numerical experiments. In particular we demonstrate improved scalability compared to to state-of-the-art rank-structured approaches in spatial dimension $d>1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-04T03:39:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaussian process regression",
      "common non-stationary kernels",
      "equispaced fourier gaussian process",
      "log-linear scaling",
      "exploits non-uniform fast fourier transforms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}