多模式张量分解：扩展分析工具与应用

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本文档深入探讨了"Tensor Decompositions and Applications"这一主题，它着重于多维数据处理领域，特别是在光谱成像（HSI）中的广泛应用。随着多传感器技术的普及和大数据集的出现，传统的矩阵模型已显示出局限性，无法满足复杂数据的分析需求。文章指出，高阶张量（即多方式数组，N-way arrays with N≥3）作为一种多线性代数的强大工具，为数据分析提供了一种根本性的转变，使得模型更为灵活且在理论上更具有优势。张量分解是核心概念，其中包括两种特别重要的扩展自矩阵的分解方法：CANDECOMP/PARAFAC（CP）分解和Tucker分解。CP分解将一个高阶张量分解为一系列具有单位秩的简单张量之和，这类似于矩阵的奇异值分解（SVD），但适用于多维度数据。而Tucker分解则是主成分分析（PCA）的高阶版本，它揭示数据中的潜在结构，通过多模式表示提供了更丰富的信息。除了CP和Tucker分解，文档还提及其他几种张量分解方法，如INDSCAL、PARAFAC2、CANDELINC、DEDICOM以及PARATUCK2，这些方法都针对不同场景下的数据分析需求，提供了不同的解析框架。非负张量分解作为特殊变体，强调在保持数据正性的前提下进行分解，适用于那些数据元素非负或具有物理意义的场景。软件支持方面，文中提到了几个重要的工具包，如N-way Toolbox、Tensor Toolbox和Multilinear Engine，这些都是实现张量分解算法并处理大规模数据的重要工具。这些软件为实际应用提供了便利，使得科研人员和工程师能够有效地利用张量分解技术来处理和理解复杂的高维数据，从而在心理测量、化学计量学、信号处理、计算机视觉、数值分析、数据挖掘、神经科学和图分析等领域取得突破性进展。总结来说，这篇综述文章不仅介绍了张量分解的基本原理和方法，还强调了其在多学科中的实际应用价值，以及相关的软件资源，对于理解并有效利用张量技术进行现代数据分析具有重要意义。

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.

Downloaded 11/09/17 to 117.32.153.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

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

Downloaded 11/09/17 to 117.32.153.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

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