高阶张量分解：应用与软件综述

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"《Tensor Decompositions and Applications》是一篇全面探讨高阶张量分解及其应用的综述文章。在信息技术领域，尤其是数据科学和机器学习中，张量（多维数组，N-way array）作为一种关键的数据结构，其分解具有广泛的应用价值。高阶张量分解（N-way数组，其中N通常大于或等于3）在多个领域展现了重要作用，例如： 1. 心理测量学与化学计量学：通过高阶张量分解可以处理复杂的多变量关系，如多元智力分析中的因素分析。 2. 信号处理：在视频和音频信号分析中，CP分解（CANDECOMP/PARAFAC）和Tucker分解被用于提取关键特征和降维。 3. 数值线性代数：Tucker分解扩展了矩阵奇异值分解的概念，提供了对大型数据集的高效表示和处理方法。 4. 计算机视觉：在图像和视频数据的特征提取、分类和识别中，张量分解技术发挥着关键作用。 5. 数据挖掘：通过非负张量分解，如非负CANDECOMP/PARAFAC（NNCP）等，有助于发现潜在的结构和模式。 6. 神经科学：神经活动数据的分析常依赖于张量模型，以理解大脑功能的复杂交互。 7. 图分析：高阶张量可以用来表示网络结构和关系，从而支持社交网络分析和社区检测。 8. 其他领域：包括但不限于机器学习中的特征工程、深度学习中的卷积神经网络等，张量分解都是不可或缺的工具。目前，市面上有多种软件包可供研究者和工程师使用，如N-way Toolbox、Tensor Toolbox以及Multilinear Engine，它们提供了强大的工具来处理和分析高阶张量数据。这些软件包不仅支持基础的张量操作，还包含了各种分解算法的实现，以及对非负张量分解的优化。了解和掌握高阶张量分解是现代数据分析和技术发展的重要组成部分，对于科研人员和实际应用开发者来说，深入理解和运用这些技术具有显著的价值。"

462 TAMARA G. KOLDA AND BRETT W. BADER

The Khatri–Rao product [200] is the “matching columnwise” Kronecker product.

Given matrices A ∈ R

I×K

and B ∈ R

J×K

, their Khatri–Rao product is denoted by

A  B. The result is a matrix of size (IJ) ×K deﬁned by

A  B =



⊗ b

··· a

⊗ b



If a and b are vectors, then the Khatri–Rao and Kronecker products are identical,

i.e., a ⊗ b = a  b.

The Hadamard product is the elementwise matrix product. Given matrices A and

B,bothofsizeI × J, their Hadamard product is denoted by A ∗ B. The result is

also of size I × J and deﬁned by

A ∗ B =







··· a







These matrix products have properties [227, 200] that will prove useful in our

discussions:

(A ⊗ B)(C ⊗ D)=AC ⊗ BD,

(A ⊗ B)

†

= A

†

⊗ B

†

A  B C =(A  B)  C = A  (B  C),

(A  B)

(A  B)=A

A ∗ B

(A  B)

†

=((A

A) ∗ (B

B))

†

(A  B)

.(2.2)

Here A

†

denotes the Moore–Penrose pseudoinverse of A [84].

As an example of the utility of the Kronecker product, consider the following.

Let X ∈ R

×I

×···×I

and A

(n)

∈ R

×I

for all n ∈{1,...,N}. Then, for any

n ∈{1,...,N},wehave

Y = X ×

(1)

(2)

···×

(N)

⇔

(n)

= A

(n)



(N)

⊗···⊗A

(n+1)

⊗ A

(n−1)

⊗···⊗A

(1)



See [134] for a proof of this property.

3. Tensor Rank and the CANDECOMP/PARAFAC Decomposition. In 1927,

Hitchcock [105, 106] proposed the idea of the polyadic form of a tensor, i.e., express-

ing a tensor as the sum of a ﬁnite number of rank-one tensors; and in 1944 Cattell

[40, 41] proposed ideas for parallel proportional analysis and the idea of multiple axes

for analysis (circumstances, objects, and features). The concept ﬁnally became popu-

lar after its third introduction, in 1970 to the psychometrics community, in the form of

CANDECOMP (canonical decomposition) by Carroll and Chang [38] and PARAFAC

(parallel factors) by Harshman [90]. We refer to the CANDECOMP/PARAFAC de-

composition as CP, per Kiers [122]. M¨ocks [166] independently discovered CP in the

context of brain imaging and called it the topographic components model. Table 3.1

summarizes the diﬀerent names for the CP decomposition.

464 TAMARA G. KOLDA AND BRETT W. BADER

It is often useful to assume that the columns of A, B,andC are normalized to length

one with the weights absorbed into the vector λ ∈ R

so that

(3.3) X ≈ λ ; A, B, C ≡



r=1

◦ b

◦ c

We have focused on the three-way case because it is widely applicable and suf-

ﬁcient for many needs. For a general Nth-order tensor, X ∈ R

×I

×···×I

,theCP

decomposition is

X ≈ λ ; A

(1)

, A

(2)

,...,A

(N)

 ≡



r=1

(1)

◦ a

(2)

◦···◦a

(N)

where λ ∈ R

and A

(n)

∈ R

×R

for n =1,...,N. In this case, the mode-n matri-

cized version is given by

(n)

≈ A

(n)

Λ(A

(N)

···A

(n+1)

 A

(n−1)

···A

(1)

)

where Λ = diag(λ).

3.1. Tensor Rank. The rank of a tensor X, denoted rank(X), is deﬁned as the

smallest number of rank-one tensors (see section 2.1) that generate X as their sum

[105, 141]. In other words, this is the smallest number of components in an exact CP

decomposition, where “exact” means that there is equality in (3.1). Hitchcock [105]

ﬁrst proposed this deﬁnition of rank in 1927, and Kruskal [141] did so independently

50 years later. For the perspective of tensor rank from an algebraic complexity point

of view, see [107, 37, 130, 24] and references therein. An exact CP decomposition

with R =rank(X) components is called the rank decomposition.

The deﬁnition of tensor rank is an exact analogue to the deﬁnition of matrix rank,

but the properties of matrix and tensor ranks are quite diﬀerent. One diﬀerence is

that the rank of a real-valued tensor may actually be diﬀerent over R and C.For

example, let X be a tensor whose frontal slices are deﬁned by





and X



−10



This is a tensor of rank three over R and rank two over C . The rank decomposition

over R is X = A, B, C,where

A =



10 1

01−1



, B =



101

011



, and C =



110

−111



whereas the rank decomposition over C has the following factor matrices instead:

A =

√



−ii



, B =

√



i −i



, and C =



i −i



This example comes from Kruskal [142], and the proof that it is rank three over R

and the methodology for computing the factors can be found in Ten Berge [212]. See

also Kruskal [143] for further discussion.

Another major diﬀerence between matrix and tensor rank is that (except in special

cases such as the example above) there is no straightforward algorithm to determine

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