多维数据分析：无监督方法与应用综述

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"Unsupervised Multiway Data Analysis: A Literature Survey" 这篇文献综述由Evrim Acar和Bu¨lent Yener撰写，探讨了多维数据分析在无监督学习中的应用。传统的两维数据分析方法（如矩阵分析）往往不足以揭示多模式数据集的全部信息内容和潜在结构。随着多维数据集在各个领域的广泛出现，包括化学计量学、神经科学、社会网络分析、文本挖掘和计算机视觉，多维数据分析作为一种探索性分析工具，其重要性日益凸显。多维数据分析（Multiway Data Analysis）是将两维数据分析扩展到更高维度的数据集。这种方法的历史可以追溯到20世纪20年代Hitchcock对张量分解的研究。文章中，作者们回顾了在多维模型、算法以及它们在不同领域应用中的显著贡献，这些领域包括但不限于： 1. **化学计量学（Chemometrics）**：在化学分析中，多维数据可能来自于不同条件下的光谱或色谱数据，通过多维分析可以发现化学物质的复杂相互作用和模式。 2. **神经科学（Neuroscience）**：在大脑成像研究中，多模态数据（如fMRI和EEG）可以揭示大脑功能和结构的复杂网络，多维分析有助于理解和解析这些复杂数据。 3. **社会网络分析（Social Network Analysis）**：在社交网络中，个体之间的互动和关系形成多维数据，多维分析能够揭示社交网络的群体结构和动态变化。 4. **文本挖掘（Text Mining）**：文本数据通常包含多个维度，如词汇、主题和情感，多维分析可以帮助识别隐藏的主题和模式，提高信息提取的效率。 5. **计算机视觉（Computer Vision）**：图像和视频数据是典型的高维数据，多维分析可以用于特征提取、目标检测和场景理解等任务。文章中讨论的关键技术之一是**高阶奇异值分解（Higher-Order Singular Value Decomposition, HOSVD）**，这是多线性代数的一个重要工具，用于分解高维数据并提取核心结构。此外，多维数据分析还包括其他方法，如多线性主成分分析（Multilinear Principal Component Analysis, MPCA）、 Tucker分解和 CANDECOMP/PARAFAC (CP) 分解等。通过这些技术，研究人员和分析师能够揭示多模态数据集中的非线性结构和模式，这对于理解和解释复杂数据至关重要。然而，多维数据处理也面临着计算复杂性和解释性挑战，因此，文献综述还可能涵盖了算法优化、可解释性方法以及在实际应用中如何克服这些挑战的案例研究。这篇综述对于希望深入理解和应用多维数据分析方法的学者和实践者来说，是一份宝贵的资源，它总结了这一领域的关键进展，并为未来的研究提供了方向。

On the other hand, if the three-way array is modeled using

an R-component PARAFAC model, we need to determine

only ðI

þ I

ÞR parameters. In that sense, a multiway

model can be considered statistically simpler than a two-way

model [16], [17].

Multilinear models, i.e., PARAFAC [18], Tucker [8], [9],

[19], and their derivatives, capture the multilinear structure

in data. Multilinearity of the model denotes that the model

is linear in each mode, and factors extracted from each mode

are linear combinations of the variables in that mode. A

component matrix, whose columns are the factors deter-

mined by the model, is then constructed to summarize the

structure in each mode. These models have been applied on

various data sets shown to contain multilinear structure,

e.g., three-way fluorescence spectroscopic data sets with

modes: samples, emission, and excitation wavelengths [20],

wavelet-transformed multichannel EEG arranged as a three-

way array with modes: frequency, time samples, and

channels [3], [21], [22], and on many more data types

described briefly in this survey.

We categorize multiway models and study their simila-

rities and differences in Section 2. In Section 3, we give an

overview of the algorithms used in fitting these models to

multiway data sets. Multiway data analysis applications

from various fields and available software tools are

summarized in Sections 4 and 5. After a brief summary of

the survey in Section 6, future research directions, particu-

larly in computer science, are discussed in Section 7.

2MULTIWAY MODELS

The most well-known multiway models in literature are

Tucker models and the PARAFAC model, which is also

called Canonical Decomposition (CANDECOMP) [23].

will briefly describe these models as well as recent models

built on the principles of PARAFAC and Tucker. This

survey studies multiway models under three categories

given in Fig. 4. First category describes PARAFAC and some

other models, which have relaxed the restrictions enforced

by a PARAFAC model to capture data-specific structures.

Second category contains the models that belong to the

Tucker family and the extensions of Tucker models. Last

category includes the models, which fall under neither the

first nor the second category but still address the problem of

analyzing multiway arrays. In spite of the categorization,

models in different families are closely related to each other,

e.g., PARALIND can be considered as a constrained version

of a Tucker3 model. This categorization is primarily for the

ease of presentation and understanding of the models. In the

rest of this paper, we discuss these models in the context of

three-way arrays but most of these models (see Tables 1

and 2) have already been extended to N-way arrays.

2.1 PARAFAC Family

2.1.1 PARAFAC

PARAFAC [18], which has been originally introduced as

the polyadic form of a tensor in [1], is an extension of

bilinear factor models to multilinear data. PARAFAC is

based on Cattell’s principle of Parallel Proportional

Profiles [24]. The idea behind Parallel Proportional Profiles

is that if the same factors are present in two samples under

8 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. 21, NO. 1, JANUARY 2009

1. CANDECOMP was proposed independently and considered equiva-

lent to PARAFAC.

Fig. 4. The categorization of multiway models studied in this survey. We study multiway models under three categories: PARAFAC family, Tucker

family, and alternative approaches.

TABLE 1

Selection of Models from PARAFAC Family



PARAFAC2 and PARALIND are expressed in matrix notation to make it easier to compare with PARAFAC representation in (3). In PARALIND, H

represents the dependency matrix. For Shifted PARAFAC, s

represents the shift at column j for the rth factor. In Convolutive PARAFAC,  is used

to capture the shifts in the log-frequency spectrogram.

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