极限学习机与自适应稀疏表示：图像分类的高效融合

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本文主要探讨了极限学习机(Extreme Learning Machine, ELM)和自适应稀疏表示(Adaptive Sparse Representation, ASR)在图像分类领域的结合应用。这两种方法在近年来的研究中都展现出了显著的优势：ELM以其快速的学习速度而备受瞩目，而SRC则因能够实现高精度分类而受到重视。极限学习机作为一种基于神经网络的快速训练算法，其核心特征在于隐藏层神经元的权重在整个训练过程中保持固定不变，仅通过调整输入和输出层之间的连接权重即可完成模型训练。这大大简化了传统反向传播法的计算负担，使得ELM在处理大规模数据集时具有明显的时间优势。然而，ELM的简单结构可能导致泛化性能有所欠缺，需要与更复杂的表示学习技术相结合以提高分类效果。另一方面，自适应稀疏表示是一种利用稀疏编码来提取图像特征的方法。通过寻找最少数目的基向量来近似输入样本，ASR能够在高维数据集中捕捉到关键的特征信息。这种表示方式不仅有助于降低维度，还可以提高分类的鲁棒性，因为它对噪声和异常值具有一定的抵抗能力。本文将这两种方法结合起来，提出了一种新型的图像分类策略。首先，使用ELM作为预训练阶段，通过其快速收敛性获取初始的图像特征表示。然后，这些特征被送入自适应稀疏表示模块，进一步优化并寻找最佳的稀疏表示，以提升分类的准确性。这种方法巧妙地结合了ELM的高效性和SRC的精确度，通过leave-one-out交叉验证(Leave-One-Out Cross Validation, LOOCV)来评估模型的性能，确保了在保证速度的同时，也实现了较高的分类精度。本文的研究为图像分类任务提供了一个有效的解决方案，展示了极限学习机和自适应稀疏表示结合的优势，这对于提升图像识别系统的实时性和准确性具有重要的实践价值。未来的研究可能着重于进一步优化算法，使其在更多类型的图像分类任务中取得更好的表现，并且探索如何在实际应用中更好地融合这两种方法的技术优势。

J. Cao et al. / Neural Networks 81 (2016) 91–102 93

Algorithm 1: ELM based classification.

Input : A training set {(x

, t

)}

j=1

, activation function g(x),

hidden node number L and a query image y.

Output: Class label of y

1 Randomly generate hidden node parameters

, b

), i = 1, 2, . . . , L.

2 Calculate H(w

, . . . , w

, x

, . . . , x

, b

, . . . , b

3 Determine the output weight matrix

β = H

4 Calculate the actual output o of ELM with respect to y.

5 Label(y) = arg max

d∈{1,...,m}

(o).

where λ is the regularization parameter to make a tradeoff

between the training error term and regularization term. The

solution of (4) is given as follows Huang (2015):

• If L ≤ N

β = (H

H + λI)

−1

T. (5)

• If L ≥ N

β = H

(HH

+ λI)

−1

T (6)

where I is the identity matrix.

2.2. SRC

SRC algorithm first estimates the sparse representation coef-

ficients of a query sample through an over-complete dictionary

whose basis atoms are extracted features with known labels. Then,

the classification is performed by finding a minimum residual error

based on the sparse coefficient (Cao, Zhao, Lai, Chen et al., 2015; Liu,

Guo, & Sun, 2016; Liu, Liu, & Sun, 2015; Wright, Yang et al., 2009;

Yang et al., 2010, 2013). Particularly, SRC algorithm assumes that

the query image can be well approximated by a linear combination

of a few basis elements.

Assuming an image database {A

}

d=1

for m classes, A

∈ R

n×k

contains k

training images belonging to the dth class, with each

image feature being concatenated as a column vector. The coding

dictionary is constructed as A = [A

, A

, . . . , A

]. For a query

image y (a column vector containing features of an image), the

objective is to derive the optimal sparse representation coefficient

via the following optimization

x = arg min

∥x∥

s.t. Ax = y (7)

where ∥·∥

is the ℓ

-norm, denoting the support of a vector. Owing

to the NP-hard problem of (7) and the progress of compressed

sensing (Donoho, 2006), researchers resort to solve the following

convex ℓ

-norm minimization problem

x = arg min

∥x∥

s.t. ∥Ax − y∥

< ε (8)

where ε is a small error tolerance. Note that numerous algorithms

with different constraints on the sparse representation coefficient

have been developed for certain applications (Babacan, Molina,

& Katsaggelos, 2010; Cao & Lin, 2014, 2015b; Wright, Nowak,

& Figueiredo, 2009; Yang et al., 2010). In SRC, the characteristic

function δ

(

x) maps to a new vector whose nonzero elements

are the entries in

x associated to the dth class. Thus, for a query

sample y belonging to the dth category, it is believed that it can

be represented by the atoms from the dth class with the minimum

residual r

(y) = ∥y − Aδ

(

x)∥

. In other words, the classification

can be achieved by finding the minimum residual as

Label(y) = arg min

d∈{1,2,...,m}

(y) with r

(y) = ∥y − Aδ

(

x)∥

. (9)

Algorithm 2: SRC based classification.

Input : A dictionary with m classes A = [A

, A

, . . . , A

], a

query image y

Output: Class label of y

1 Normalize the columns of A to have unit l

-norm.

2 Solve the optimization problem (8).

3 Calculate the residuals

(y) = ∥y − A

(

x)∥

, d = 1, 2, . . . , m.

4 Label(y) = arg min

d∈{1,2,...,m}

(y).

Algorithm 2 summarizes the SRC based classification.

SRC has shown various promising features in face recognition

and image processing (Bai et al., 2015; Cao, Zhao, Lai, Chen et al.,

2015; Cao, Zhao, Lai, Ong et al., 2015; Luo & Zhang, 2014; Wang

et al., 2014; Wright, Yang et al., 2009; Zhu et al., 2016). First of all, in

contrast with most of the existing classifiers whose performances

highly depend on feature extraction, it is pointed out in Wright,

Yang et al. (2009) that SRC with accurate sparse coefficient opti-

mization and a complete dictionary with enough features is not

sensitive to feature extraction. The second important merit of SRC

is that it is training-free. Despite this, the SRC algorithm suffers the

long testing time issue, which is incurred by the ℓ

-norm mini-

mization problem (Cao, Zhao, Lai, Chen et al., 2015; Cao, Zhao, Lai,

Ong et al., 2015; Luo & Zhang, 2014). Experiments in Cao, Zhao,

Lai, Chen et al. (2015), Cao, Zhao, Lai, Ong et al. (2015) and Luo

and Zhang (2014) have shown that the sparse representation coef-

ficient estimation for some algorithms takes a few seconds or even

dozens of seconds per sample. Such a long testing time has severely

limited the applicable ranges of SRC, especially in the current big

data era.

To reduce the overall computation burden, the cascade classifier

ELM-SRC has been developed for image classification (Cao, Zhao,

Lai, Ong et al., 2015; Luo & Zhang, 2014). Incorporating with the

fast training and testing property of ELM and the good image

classification capability of SRC, the hybrid ELM-SRC classifier

speeds up the overall classification procedure through partitioning

the testing image dataset into two sub-sets and adaptively

choosing ELM and SRC to perform classification. Although ELM-SRC

has an advantage over ELM in classification accuracy and shows

superiority over SRC in testing speed, its classification accuracy is

generally not as good as SRC. In addition, ELM-SRC still has room for

improvement due to using the over-complete or highly redundant

dictionary for linear representation.

3. The proposed EA-SRC

To further reduce the misclassification rate as well as the com-

putational complexity of the hybrid ELM-SRC classifier, we de-

velop a novel image classification algorithm based on the regular-

ized ELM with an optimal regularization parameter and SRC with

an adaptive sub-dictionary selection strategy. The contributions

of this paper are two-fold: (1) the LOO cross validation approach

is adopted to optimize the regularization parameter of ELM and

thus the best classification hyperplane is achieved; (2) for each

query image, an adaptive sub-dictionary selection strategy using

the output vector derived by the optimized ELM is developed for

sparse representation dictionary construction. Details of the pro-

posed classifier are given in the following.

3.1. LOO error based optimization

The leave-one-out cross validation (LOO) approach is one of

the most effective methods for model selection and parameter

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