张量最大余量准则：一种增强的监督降维方法

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本文主要探讨了"具有张量表示的最大余量准则" (Tensor-based Maximum Margin Criterion, TMMC) 这一新颖的机器学习技术，特别应用于有监督降维领域。在传统的最大余量准则 (Maximum Margin Criterion, MMC) 主要处理向量数据的基础上，TMMC向前迈进了一步，将图像对象编码为n阶张量，这种多维度的数据结构允许更丰富的信息表达。张量的2D表示直接被转换为矩阵，这在数学运算上更加直观和高效。 TMMC的核心在于利用k模式优化方法对高阶张量进行处理。通过迭代学习多个相互关联的判别子空间，这种方法能够有效地降低张量的维数，从而实现数据的有效压缩和特征提取。这种方法不仅保留了原始数据的主要信息，而且能够在保持分类性能的同时，减少计算复杂性和存储需求。实验结果在四个不同的数据库上展现了TMMC的优势。与并行子空间分析 (CSA) 方法相比，TMMC在准确识别率上表现优秀，显示出在处理大规模、高维度图像数据时的优越性。然而，值得注意的是，尽管基于张量和矩阵的数据表示方法在某些情况下表现出色，但实验也揭示了一个现象：并非所有情况下它们的识别精度都高于基于矢量的方法。这可能是由于张量表示的复杂性带来的额外挑战，比如过拟合或对数据分布的敏感性。本文作者对这一现象进行了深入的分析和讨论，指出这可能与张量的特性、模型选择以及数据预处理等因素有关。他们强调了在实际应用中需要根据具体任务和数据特性来选择最合适的降维策略，而不是盲目追求更高的维度转换效果。TMMC的提出不仅扩展了MMC理论的应用范围，也为后续的研究提供了新的视角，即如何更有效地结合张量数据结构进行无损或近似无损的降维和特征学习。

Maximum margin criterion with tensor representation

Rong-Xiang Hu

a,b

, Wei Jia



, De-Shuang Huang

, Ying-Ke Lei

a,b,c

Hefei Institute of Intelligent Machines, Chinese Academy of Science, PO Box 1130, Hefei 230031, China

Department of Automation, University of Science and Technology of China, Hefei 230027, China

Department of Information, Electronic Engineering Institute, Hefei 230037, China

article info

Keywords:

Tensor representation

Maximum Margin Criterion

Subspace learning

abstract

In this paper, we propose tensor based Maximum Margin Criterion algorithm (TMMC) for supervised

dimensionality reduction. In TMMC, an image object is encoded as an nth-order tensor, and its 2-D

representation is directly treated as matrix. Meanwhile, the k-mode optimization approach is exploited

to iteratively learn multiple interrelated discriminative subspaces for dimensionality reduction of the

higher order tensor. TMMC generalizes the tradition al MMC based on vector data to the one based on

matrix and tensor data, which completes the MMC family in terms of data representation. The results of

experiments conducted on four databases show that the accurate recognition rate of TMMC is better

than that of the method of Concurrent Subspaces Analysis (CSA), and is comparable with the method of

Multilinear Discriminant Analysis (MDA). The experimental results also show that the accurate

recognition rate of the tensor/matrix-based methods may not always be better than that of vector-

based methods. Reasonable discussions about this phenomenon have been given in this paper.

1. Introduction

In recent years, the research on dimensionality reduction

[1,2,19,20,29,30] is very active, and many algorithms of dimen-

sionality reduction are mainly utilized for image recognition.

Particularly, in the past decade, these algorithms have been

through a series of changes in the representation of data. In the

early stage of the research in this ﬁeld, almost all representative

algorithms including Principal Component Analysis (PCA) [1] and

Linear Discriminant Analysis (LDA) [2] require that the 2-D image

data must be reshaped into 1-D vector, which can be referred to as

the strategy of ‘‘image-as-vector’’. However, the operation of

vectorization inevitably leads to two obvious disadvantages.

Firstly, it usually causes the ‘‘curse of dimensionality’’ dilemma

and ‘‘small sample size’’ problem. Secondly, it might break the

intrinsic 2-D structure of an image matrix, which results in the

failure of exploration of spatial/correlation information. Thus,

there is an increasing demand for uncovering the underlying

structures in different data representations to enhance the

performance of dimensionality reduction algorithms. With the

development of research, it is believed that, in the real world, an

object often has some specialized structures and such structures

are intrinsically in the form of second or even higher order tensor.

In this regard, vector is the ﬁrst-order tensor and matrix is the

second-order tensor.

When vector based PCA and LDA were extended to matrix

based PCA and LDA [3–11], this strategy is called as ‘‘image-as-

matrix’’. Generally, the matrix-based algorithms have two models,

which are unilateral projection and bilateral projection, respec-

tively. Additionally, the former can be considered as a special case

of the latter.

Yang et al. [3] proposed 2DPCA, which only used one

projection matrix to conduct dimensionality reduction with a 2-

D matrix representation, referred to as unilateral projection. Later,

Xu et al. [4] proposed Coupled Subspace Analysis (Xu et al. called

it as CSA-2), in which there are two projection matrices operating

on both sides of the image matrix, referred to as bilateral

projection. Ye [5] proposed Generalized Low Rank Approxima-

tions of Matrices (GLRAM), which is nearly identical to CSA-2.

Kong et al. [6] proposed Bilateral 2DPCA (B2DPCA), which is the

generalized version of 2DPCA by bilateral projection. In [7],

TensorPCA (TPCA) was proposed by Cai et al. However, although

the word ‘tensor’ was adopted, TPCA only exploited the

representation of matrix. And, it belongs to the model of bilateral

projection. Generally speaking, all of the methods introduced in

this paragraph could be regarded as the extended PCA based on

the strategy of ‘‘image-as-matrix’’.

After Yang’s 2DPCA was proposed, Kong et al. [8] proposed 2D

Fisher discriminant analysis (2DFDA), and Li and Yuan [9]

proposed 2DLDA [9] (we call it as 2DLDA1), both of which have

similar strategy with 2DPCA, and belong to the model of unilateral

projection. Later, Ye et al. [10] proposed another form of 2DLDA

(we call it as 2DLDA2), which simultaneously computed two

subspaces. TensorLDA (TLDA) proposed by Cai et al., utilized the

ARTICLE IN PRESS

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/neucom

Neurocomputing

doi:10.1016/j.neucom.2009.11.036



Corresponding author at: Hefei Institute of Intelligent Machines, Chinese

Academy of Science, PO Box 1130, Hefei 230031, China.

E-mail address: icg.jiawei@gmail.com (W. Jia).

Neurocomputing 73 (2010) 1541–1549

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