---
tag: Z855M8QN2H
kind: definition
name: Tensor decomposition
msc: [15A69, 15A23, 65F55]
defines: VWV7TA2AJX
uses: [000AYPA9AJ, 0BF2DE2SYK, W84NTF77M0]
sources:
  - url: https://en.wikipedia.org/wiki/Tensor_decomposition
---

## Definition

Let $F$ 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]]).
A *tensor decomposition* of $T$ is an expression of $T$ in one of the following standard forms.

1. A *canonical polyadic (CP) decomposition* of length $r$ is an identity
$$T = \sum_{i=1}^{r} a_i^{(1)} \otimes a_i^{(2)} \otimes \cdots \otimes a_i^{(d)}, \qquad a_i^{(k)} \in F^{n_k}.$$
2. A *Tucker decomposition* with multilinear size $(r_1, \ldots, r_d)$ is an identity
$$T = \sum_{j_1 = 1}^{r_1} \cdots \sum_{j_d = 1}^{r_d} g_{j_1 \ldots j_d} \, u_{j_1}^{(1)} \otimes \cdots \otimes u_{j_d}^{(d)},$$
where $G = (g_{j_1 \ldots j_d})$ is a core tensor of size $r_1 \times \cdots \times r_d$ and $u_j^{(k)} \in F^{n_k}$.

## Remarks

The smallest length of a CP decomposition of $T$ is the [[tensor rank|W84NTF77M0]] of $T$.
A CP decomposition is a Tucker decomposition whose core is diagonal, with $r_1 = \cdots = r_d = r$.
Example: 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 $2$.
Check: the first term has only the entry $(1,1,1)$ equal to $1$, and the second only the entry $(2,2,2)$.
Non-example: $e_1 \otimes e_1 \otimes e_1$ alone is not a CP decomposition of this array, since it misses the entry $t_{222} = 1$.
For $d = 2$, 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.
For $d = 3$, Kruskal's theorem gives conditions under which a CP decomposition of minimal length is essentially unique.
Here essentially unique means unique up to the order of the terms and the scaling of the factors.
This 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.