L1-2DCRP：基于L1范数的降维与人脸识别的稳健投影

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在信号处理和图像通信领域，研究者们一直在寻求提高数据处理效率和鲁棒性的方法。本文介绍了一种新颖的降维技术——基于L1范数的二维协作表示的稳健投影（L1-2DCRP）。传统的协作表示投影（CRP）基于L2范数，虽然在识别和计算机视觉任务中表现优异，但它对噪声和异常值敏感，并且在处理高维图像时容易受到维度灾难的影响。 L1-2DCRP的主要创新在于它采用L1范数作为优化目标，相比于L2范数，L1范数具有更好的抗噪性和稀疏性。L1范数对于异常值不那么敏感，因为它倾向于产生更为稀疏的表示，有助于减少噪声带来的干扰。这意味着在进行人脸识别或图像特征提取时，L1-2DCRP能够提供更稳定和有效的特征映射，即使在数据集中存在噪声和异常样本的情况下也能保持较好的性能。在实现L1-2DCRP时，研究者们可能采用了迭代优化算法，如拉格朗日乘子法或高效算法，如FISTA（Fast Iterative Shrinkage-Thresholding Algorithm），来求解L1范数下的最优协作表示。这种方法可以显著降低原始高维数据的复杂性，同时保留关键信息，这对于大规模数据集的分析和处理至关重要。此外，文章还提到了关键词“Collaborative Representation-based Projection”（协作表示投影）、“L1-2DCRP”以及“Dimensionality Reduction”，这些是理解论文核心内容的关键术语。L1-2DCRP不仅关注提升模型的鲁棒性，而且致力于在降低维度的同时保持特征间的内在关联，这对于实际应用中的特征选择和降维任务具有重要的实际意义。这篇论文提出了一种改进的CRP方法，旨在解决L2范数下存在的问题，通过引入L1范数增强了模型的稳健性和对高维图像数据的有效处理能力。这是一项对于提升信号处理和图像通信领域数据处理效率、特别是面对噪声和复杂环境时的降维技术的重要贡献。

Signal Processing: Image Communication 81 (2020) 115684

Contents lists available at ScienceDirect

Signal Processing: Image Communication

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

Robust L1-norm two-dimensional collaborative representation-based

projection for dimensionality reduction

✩

Lulu He, Jimin Ye

∗

, Jianwei E

School of Mathematics and Statistics, Xidian University, Xi’an 710071, China

A R T I C L E I N F O

Keywords:

Collaborative representation-based projection

(CRP)

L1-2DCRP

L1-norm

Face recognition

Dimensionality reduction

A B S T R A C T

Collaborative representation-based projection (CRP) is a well-known dimensionality reduction technique, which

has been proved to have better performance than sparse representation-based projection (SRP) in the fields of

recognition and computer vision. However, classical CRP is sensitive to noises and outliers since its objective

function is based on L2-norm, and it will suffer from the curse of dimensionality as it is used for images

processing. In this paper, a novel CRP model, named L1-norm two-dimensional collaborative representation-

based projection (L1-2DCRP) and an efficient iterative algorithm to solve it are proposed. Different from

conventional CRP, the optimal problem in our proposed model is a L1-norm-based maximization and the

vector data is extended to matrix date. The proposed algorithm is theoretically proved to be monotonously

convergent, and more robust to noises and outliers since L1-norm is used. Experimental results on CMU Multi-

PIE, COIL20, FERET and ORL face databases validate the effectiveness of L1-2DCRP compared with several

state-of-the-art approaches.

1. Introduction

Dimensionality is always a serious problem in the fields of data

analysis and pattern recognition because data are always with high

dimensionality in the real world. In data analysis processing, high-

dimensional data need more training time and storage space, on which

it is not realistic to perform analysis directly. Dimensionality reduction

is the key step used to reduce the dimension of input space to simplify

data analysis problem in this field. In the last few decades, various

dimensionality reduction techniques haven been developed [1]. Among

these methods, principal component analysis (PCA) [2] and linear

discriminant analysis (LDA) [3] are two of the most popular ones. PCA

is currently the most famous and widely used linear dimensionality

reduction algorithm and its core idea is to project data along the direc-

tion of maximum variance of samples in the feature space to minimize

reconstruction error. However, PCA is an unsupervised method. LDA

algorithm can make full use of the label information and with the

idea of maximizing the Fisher discriminant criterion to seek optimal

projection vectors. As we all know, LDA is superior to PCA since LDA

owns more discriminant ability.

Recently, sparse representation (SR) has gained widely attention

and has been successfully applied to solve many practical problems [4–

6]. SR-based methods construct sparsity L1-graph, which is robustness

to outliers and can be divided into unsupervised and supervised. Qiao

✩

No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work.

For full disclosure statements refer to https://doi.org/10.1016/j.image.2019.115684.

∗

Corresponding author.

E-mail address: jmye@mail.xidian.edu.cn (J. Ye).

et al. [7] proposed sparsity preserving projections (SPP), in which the

sparse reconstructive relationship of data was preserved by minimizing

a L1 regularization-related objective function. Chen et al. [8] presented

sparse neighborhood preserving embedding (SNPE), which is identical

to SPP to some extent. While both of the two received good perfor-

mance in data classification, they are unsupervised. In order to further

improve the performance of SPP and SNPE, Gui et al. [9] and Zhang

et al. [10] embedded discriminant information into these two methods

and provided the discriminant sparse neighborhood preserving embed-

ding (DSNPE) and sparse locality preserving discriminative projections

(SLPDP), respectively.

The aforementioned approaches are all 1D vector-based algorithms,

in which 2D matrix data need to be converted into 1D vector form. As

we all know, data in real wold are always with high dimensionality and

the transformation will cause the so-called ‘‘curse-of-dimensionality’’.

In addition, the transformation will destroy the inner local charac-

teristics hidden in data. To address this shortcomings, several 2D

matrix-based methods have been developed. Yong et al. [11] incor-

porated the 2D matrix into PCA and proposed 2DPCA, in which the

covariance matrix was constructed directed by 2D matrix data. Ye

et al. [12] maximized the ratio of 2D between-class scatter matrix and

the 2D within-class scatter matrix and proposed 2DLDA. Compared

with PCA and LDA, 2DPCA and 2DLDA can achieve better performance

https://doi.org/10.1016/j.image.2019.115684

Received 10 June 2019; Received in revised form 29 October 2019; Accepted 1 November 2019

Available online 8 November 2019

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