{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $F$ be a field, let $d \\ge 2$, and let $T \\in F^{n_1} \\otimes \\cdots \\otimes F^{n_d}$ be a tensor, that is, an array of size $n_1 \\times \\cdots \\times n_d$ (see [[tensor product|0BF2DE2SYK]]).\nA *tensor decomposition* of $T$ is an expression of $T$ in one of the following standard forms.\n\n1. A *canonical polyadic (CP) decomposition* of length $r$ is an identity\n$$T = \\sum_{i=1}^{r} a_i^{(1)} \\otimes a_i^{(2)} \\otimes \\cdots \\otimes a_i^{(d)}, \\qquad a_i^{(k)} \\in F^{n_k}.$$\n2. A *Tucker decomposition* with multilinear size $(r_1, \\ldots, r_d)$ is an identity\n$$T = \\sum_{j_1 = 1}^{r_1} \\cdots \\sum_{j_d = 1}^{r_d} g_{j_1 \\ldots j_d} \\, u_{j_1}^{(1)} \\otimes \\cdots \\otimes u_{j_d}^{(d)},$$\nwhere $G = (g_{j_1 \\ldots j_d})$ is a core tensor of size $r_1 \\times \\cdots \\times r_d$ and $u_j^{(k)} \\in F^{n_k}$.\n\n## Remarks\n\nThe smallest length of a CP decomposition of $T$ is the [[tensor rank|W84NTF77M0]] of $T$.\nA CP decomposition is a Tucker decomposition whose core is diagonal, with $r_1 = \\cdots = r_d = r$.\nExample: the $2 \\times 2 \\times 2$ array with $t_{111} = t_{222} = 1$ and all other entries $0$ has the CP decomposition $e_1 \\otimes e_1 \\otimes e_1 + e_2 \\otimes e_2 \\otimes e_2$ of length $2$.\nCheck: the first term has only the entry $(1,1,1)$ equal to $1$, and the second only the entry $(2,2,2)$.\nNon-example: $e_1 \\otimes e_1 \\otimes e_1$ alone is not a CP decomposition of this array, since it misses the entry $t_{222} = 1$.\nFor $d = 2$, a CP decomposition is a factorization $T = A B^T$ with $r$ columns in $A$ and $B$, and the [[singular value decomposition|000NPTMX6J]] gives one of minimal length.\nFor $d = 3$, Kruskal's theorem gives conditions under which a CP decomposition of minimal length is essentially unique.\nHere essentially unique means unique up to the order of the terms and the scaling of the factors.\nThis has no analogue for matrices: a minimal factorization $T = A B^T$ of a matrix of rank at least $2$ is never unique in this sense.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "VWV7TA2AJX",
    "kind": "definition",
    "msc": [
      "15A69",
      "15A23",
      "65F55"
    ],
    "name": "Tensor decomposition",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Tensor_decomposition"
      }
    ],
    "tag": "Z855M8QN2H",
    "uses": [
      "000AYPA9AJ",
      "0BF2DE2SYK",
      "W84NTF77M0"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/Z855M8QN2H/",
  "links": [
    {
      "field": "defines",
      "target": "VWV7TA2AJX"
    },
    {
      "field": "uses",
      "target": "000AYPA9AJ"
    },
    {
      "field": "uses",
      "target": "0BF2DE2SYK"
    },
    {
      "field": "uses",
      "target": "W84NTF77M0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Z855M8QN2H.md",
  "msc": [
    "15A69",
    "15A23",
    "65F55"
  ],
  "name": "Tensor decomposition",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Z85/Z855M8QN2H.md",
  "status": null,
  "tag": "Z855M8QN2H",
  "uri": "m2://Z855M8QN2H",
  "url": "https://math2.tamnd.com/wiki/tensor-decomposition-definition/",
  "writing": {
    "tag": "Z855M8QN2H",
    "kind": "definition",
    "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
      }
    ]
  }
}