{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "77NG85NDQM"
    }
  ],
  "body": "\n## Summary\n\nMultivariate approximation is the part of [[approximation theory|YPJSH79SHQ]] that approximates functions of several variables.\nIts tools include [[polynomials in several variables|07GNX8GCYP]], tensor product methods, [[radial basis functions|XZ33P3HBB9]], box splines, [[sparse grids|84BR7X6HET]] and neural networks.\nMany one-dimensional facts fail or change form here: for instance, interpolation by polynomials at arbitrary distinct points is no longer always possible.\nA central question is how the error depends on the dimension, which leads to the [[curse of dimensionality|YS424FTY4E]].\n\n## Explanations\n\n### L1 human\n\nA function of one variable is a curve, and a function of two variables is a surface; in many applications there are dozens or thousands of variables.\nMultivariate approximation asks how to replace such a function by a simpler one, such as a polynomial, that is close to it everywhere.\nThe number of simple building blocks grows quickly with the number of variables.\nFor example, polynomials of degree at most $2$ in one variable form a space of dimension $3$, but in $10$ variables they form a space of dimension $\\binom{12}{2} = 66$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "41A63",
      "41A10"
    ],
    "name": "Multivariate approximation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Approximation_theory"
      }
    ],
    "specializes": [
      "YPJSH79SHQ"
    ],
    "tag": "CT3PNBCYMX"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/CT3PNBCYMX/",
  "links": [
    {
      "field": "specializes",
      "target": "YPJSH79SHQ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/CT3PNBCYMX.md",
  "msc": [
    "41A63",
    "41A10"
  ],
  "name": "Multivariate approximation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CT3/CT3PNBCYMX.md",
  "status": null,
  "tag": "CT3PNBCYMX",
  "uri": "m2://CT3PNBCYMX",
  "url": "https://tamnd.github.io/math2/wiki/multivariate-approximation/",
  "writing": {
    "tag": "CT3PNBCYMX",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}