增量学习方法：保邻拉普拉斯特征图的降维与特征提取

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本文主要探讨了增量学习在拉普拉斯特征映射(Laplacian Eigenmaps, LEM)中的应用，着重于解决传统非线性降维方法存在的问题。传统非线性主成分分析(NPCA)和拉普拉斯特征映射在处理大规模数据集时，由于其批处理性质，当新样本不断出现时，需要反复计算，这在计算效率上显得十分低效。为了克服这个局限，研究者提出了增量拉普拉斯特征映射(Incremental Laplacian Eigenmaps, ILEMs)算法。 ILEMs的核心思想是保留数据点之间的相邻信息，即在每次新样本加入时，仅对已有的低维表示进行微调，而不是重新构建整个模型。这种方法通过局部线性构造(Local Linear Construction, LLC)策略，有效地利用了现有数据点的局部结构，从而减少了计算复杂度。这种方法适用于实时处理高维数据流，特别适合于在线学习和动态环境下的数据分析，如社交网络分析、遥感图像处理等场景，其中数据集的更新频繁且数据量大。文章的流程可能包括以下几个步骤： 1. 引入问题：首先回顾了传统的批量拉普拉斯特征映射方法及其在大规模数据上的挑战。 2. 算法介绍：详细阐述了增量学习框架下如何构建增量拉普拉斯矩阵，并解释了如何通过保持相邻信息来更新特征向量。 3. 局部线性建模：强调了在新样本加入时如何通过邻域信息保持局部线性关系，使得更新过程更为高效。 4. 学习过程优化：讨论了可能的优化策略，如梯度下降或其他迭代方法，以最小化误差并保持低维表示的质量。 5. 性能评估：通过实验对比展示了增量拉普拉斯特征映射在处理增量数据集时，与传统方法相比在计算效率和准确性方面的优势。 6. 结论与未来工作：总结了研究结果，并提出了未来可能的研究方向，如扩展到其他增量学习任务或进一步提升算法的鲁棒性。这篇研究论文对于那些需要处理大量动态数据，且希望实时进行非线性降维和特征提取的领域具有重要意义，它提供了一种高效而有效的增量学习解决方案，显著降低了计算负担，同时保持了在处理非线性数据结构时的有效性。

Incremental Laplacian eigenmaps by preserving adjacent information between

data points

Peng Jia

, Junsong Yin

, Xinsheng Huang

, Dewen Hu

Department of Automatic Control, College of Mechatronics and Automation, National University of Defense Technology, Changsha, Hunan 410073, China

Beijing Institute of Electronic System Engineering, 100141, China

article info

Article history:

Received 13 May 2008

Received in revised form 22 June 2009

Available online xxxx

Communicated by M.A. Girolami

Keywords:

Laplacian eigenmaps

Incremental learning

Locally linear construction

Nonlinear dimensionality reduction

abstract

Traditional nonlinear manifold learning methods have achieved great success in dimensionality reduc-

tion and feature extraction, most of which are batch modes. However, if new samples are observed,

the batch methods need to be calculated repeatedly, which is computationally intensive, especially when

the number or dimension of the input samples are large. This paper presents incremental learning algo-

rithms for Laplacian eigenmaps, which computes the low-dimensional representation of data set by opti-

mally preserving local neighborhood information in a certain sense. Sub-manifold analysis algorithm

together with an alternative formulation of linear incremental method is proposed to learn the new sam-

ples incrementally. The locally linear reconstruction mechanism is introduced to update the existing

samples’ embedding results. The algorithms are easy to be implemented and the computation procedure

is simple. Simulation results testify the efﬁciency and accuracy of the proposed algorithms.

1. Introduction

In pattern recognition and machine learning, dimensionality

reduction aims to transform the high-dimensional data points in

a low-dimensional space, while retaining most of the underlying

structure in the data. It can be used to solve the curse of dimen-

sionality (Bellman, 1961), and to accomplish data visualization. Be-

sides some linear algorithms, such as principal component analysis

(PCA) (Turk and Pentland, 1991) and linear discriminant analysis

(LDA) (Belhumeur et al., 1997), manifold learning, which serves

as a nonlinear method, has attracted more and more attention

recently.

Several efﬁcient manifold learning techniques have been pro-

posed. Isometric feature mapping (ISOMAP) (Balasubramanian

et al., 2002) estimates the geodesic distances on the manifold and

uses them for projection. Locally linear embedding (LLE) (Roweis

and Saul, 2000) projects data points to a low-dimensional space

that preserves local geometric properties. Laplacian eigenmaps

(LE) (Belkin and Niyogi, 2003) uses the weighted distance between

two points as the loss function to get the dimension reduction re-

sults. Local tangent space alignment (LTSA) (Zhang and Zha, 2004)

constructs a local tangent space for each point and obtains the glo-

bal low-dimensional embedding results through afﬁne transforma-

tion of the local tangent spaces. Yan et al. (2007) present a general

formulation known as graph embedding to unify different dimen-

sionality reduction algorithms within a common framework.

All of the above algorithms have been widely applied. However,

serving as the batch methods, they require all training samples are

given in advance. When samples are observed sequentially, batch

method is computationally complex. This is because the batch

method needs to be run repeatedly once new samples are ob-

served. To overcome the problem, many researchers have been

working on incremental learning algorithms. The problem of incre-

mental learning can be stated as follows. Let X ¼½x

; x

; ...; x

 be a

data set, where x

2 R

. Assume that the low-dimensional coordi-

nate y

of x

for the ﬁrst n training samples are given. When a

new sample x

n+1

is observed, incremental learning should ﬁgure

out how to project x

n+1

in the low-dimensional space and to update

the existing samples’ low-dimensional coordinates (Liu et al.,

2006). Martin and Anil (2006) describe an incremental version of

ISOMAP: the geodesic distances are updated, and an incremental

eigen-decomposition problem is solved. Kouropteva et al. (2005)

assume the eigenvalues of the cost matrix remain the same when

a new data point arrives and the incremental learning problem of

LLE is tackled by solving a d

 d

minimization problem, where

is the dimensionality of the low-dimensional embedding space.

Bengio et al. (2004) cast MDS, ISOMAP, LLE and LE in a common

framework, in which these algorithms are seen as learning eigen-

functions of a kernel, and try to generalize the dimensionality

reduction results for the novel data points. An incremental mani-

fold learning algorithm via TSA is proposed by Liu et al. (2006)

doi:10.1016/j.patrec.2009.08.005

* Corresponding author. Fax: +86 731 84574992.

E-mail addresses: pengjia@nudt.edu.cn (P. Jia), jsyin@nudt.edu.cn (J. Yin),

dwhu@nudt.edu.cn (D. Hu).

Pattern Recognition Letters xxx (2009) xxx–xxx

Contents lists available at ScienceDirect

Pattern Recognition Letters

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

ARTICLE IN PRESS

Please cite this article in press as: Jia, P., et al. Incremental Laplacian eigenmaps by preserving adjacent information between data points. Pattern Recogni-

tion Lett. (2009), doi:10.1016/j.patrec.2009.08.005

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