自动加权低秩稀疏恢复：显著特征提取的新框架

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"强大的自动加权投影低秩和稀疏恢复，可直观显示" 本文主要探讨了在高维数据处理中的特征提取和分类问题，特别是针对现有的低秩稀疏表示模型存在的局限性。传统的低秩稀疏表示方法往往忽视了样本局部流形结构的保持，这可能导致模型性能的下降。为解决这一问题，研究者提出了一种名为归纳鲁棒自动加权低秩和稀疏表示（RALSR）的新框架，旨在有效地提取高维数据的显著特征。 RALSR模型的核心是将低秩和稀疏恢复相结合，并引入自适应的局部性保留权重，从而在保持数据结构的同时增强模型的鲁棒性。具体实现中，该模型通过最小化联合重建误差来计算自适应权重，这不仅考虑了恢复的干净数据，还考虑了显著特征。L1范数的使用有助于确保学习到的权重矩阵具有稀疏性，有利于去除噪声和不重要的特征。此外，RALSR的基础投影利用了联合的低秩和稀疏正则化，这增强了其在特征编码过程中的表现力，尤其在提取显著特征时。因此，通过RALSR计算得到的低秩稀疏特征对于后续的分类任务更加准确和有效。实验部分，包括视觉效果和数值比较，验证了RALSR在数据表示和分类任务上的优越性。文章发表在《神经网络》(Neural Networks)期刊上，进一步证明了该方法在视觉表示领域的贡献和价值。关键词：自动加权低秩和稀疏恢复、鲁棒表示、特征提取、分类 RALSR模型提供了一种创新的解决方案，通过结合低秩恢复和稀疏表示的优势，以及引入自适应权重机制，实现了对高维数据的高效处理和特征提取。这种方法对于提升数据表示的质量和分类准确性具有重要意义，尤其在面临复杂和噪声数据时。

L. Wang, B. Wang, Z. Zhang et al. / Neural Networks 117 (2019) 201–215 203

well handle the original data that usually contains noise and cor-

ruptions, IRPCA computes a low-rank projection matrix P to com-

pute the low-rank principal components by solving the following

nuclear-norm minimization based problem:

min

P,E

∥

∗

+ λ

∥

, s.t. X = PX + E, (1)

where

∥

∗

is the nuclear-norm,

∥

is the L1-norm, E = X − PX

is the sparse error, and λ is a positive factor depending on the

noise level (Bao et al., 2012; Liu et al., 2013, 2010). Suppose P

∗

is an optimal solution to the problem, the principal components

can be recovered as P

∗

X by embedding data into the underlying

subspace.

PCA-L1 improves PCA by replacing the original L2-norm by

L1-norm to measure the scatter matrix so that the method is

invariant to rotation and is also robust to noise and outliers. The

optimization problem of PCA-L1 is defined as

min



i=1



(

− x

)



, s.t. P

∈ R

n×d

, P

= I, (2)

where x

is the ith sample of X, x denotes the mean vector of all

samples, and P

= I is orthogonal constraint.

2.2. Robust LatLRR (rLatLRR) (Zhang & Lin et al., 2014)

rLatLRR is a robust formulation of LatLRR, and improves LatLRR

by performing the low-rank projection learning and low-rank

coefficients coding in the noise-removed clean data space. Specif-

ically, when given data X is noisy or corrupted, rLatLRR applies

the de-noised data X −E as the dictionary instead of original data,

leading to the following problem:

min

Z,P,E

∥

∗

∥

∗

+ λ

∥

, s.t. X − E =

(

X − E

)

Z + P

(

X − E

)

(3)

where Z is the low-rank coefficients matrix. It is clear that the

above model of rLatLRR model is non-convex, so it adopts a two-

stage strategy. That is, rLatLRR firstly applies RPCA (Candès et al.,

2011; Wright & Peng et al., 2009) to separate X into X = A

∗

. rLatLRR obtains the solution by solving the following relaxed

noiseless problem:

min

Z,P

∥

∗

∥

∗

, s.t. A

∗

= A

∗

Z + PA

∗

. (4)

3. Robust auto-weighted low-rank and sparse representation

framework by joint feature embedding

3.1. Formulation

In this section, we mainly present the objective function of

our RALSR. Given the data matrix X ∈ R

n×N

, our RALSR aims

at improving the data representation power in threefold. First,

to encode the extracted features accurately, our new formulation

regularizes the joint truncated nuclear norm based low-rank and

L1-norm based sparse constraints on the salient features and the

underlying projection P ∈ R

n×n

, inspired by I-LSPFC (Zhang et al.,

2016). Second, to make sure the learnt weights to be optimal

for data representation and avoid the tricky issue of determining

the optimal number of neighbors or kernel width, our RALSR

seamlessly integrates the low-rank and sparse coding with the

adaptive weight learning into a unified framework. Moreover,

the adaptive reconstruction weights are learnt by minimizing the

joint reconstruction errors simultaneously based on the original

data and salient features PX:

(

)

∥

X − XW

∥

PX − PXW

∥







−







∥

(5)

which is mainly added to encode the adaptive reconstruction

weights in W by simultaneously minimizing the reconstruction

errors based on the given data X and salient features PX of

samples, i.e.,

∥

X − XW

∥

PX − PXW

∥

. Thus, the adaptive

weights W

i,j

can measure the contribution of each sample x

to reconstruct x

. In other words, W

i,j

can preserve the locality

and reconstructive relationship between samples x

and x

in the

original input space and projective feature subspace spanned by

P, i.e., the larger W

i,j

, the more similar x

and x

. The L1-norm is

also regularized on the weight matrix W to make the weights

i,j

satisfy the sparse properties with larger values denoting the

neighbors of x

that play an important role for reconstructing

. As a result, the reconstruction error can be potentially mini-

mized, and minimizing J(W) can also enable the learnt projection

P to preserve the local neighborhood information of samples

adaptively.

Thus, based on the formulations of IRPCA and PCA-L1, the

intermediate problem of RALSR is defined as

min

P,W ,E

∥

+ α

∥









−







∥



+λ

∥

s.t. X = PX + E.

(6)

where

∥

is the truncated nuclear norm (Hu et al., 2013) that

is defined as the sum of min

(

n, N

)

− r minimum singular values,

that is,

∥



min

(

n, N

)

i=r+1

(

)

, where r is a nonnegative integer

and r ≤ min

(

n, N

)

. Note that the motivation of truncated nuclear

norm is detailed in Han, Leung, Huang, and So (2017) and Hu et al.

(2013), i.e., the values of largest r nonzero singular values will

not affect the rank of the matrix, so the truncated nuclear norm

mainly focuses on minimizing the sum of smallest min

(

n, N

)

− r

singular values. But the truncated norm

∥

is generally non-

convex, it is not easy to optimize it directly. To address this

issue, Hu et al. (2013) have clearly proved that the truncated

nuclear norm

∥

and the standard nuclear norm

∥

∗

have the

following relationship:

∥

∗

− max

=I, V





min

(

n, N

)



i=1

(

)

−



i=1

(

)

min

(

n, N

)



i=r+1

(

)

∥

(7)

max

=I, V







i=1

(

)

, (8)

where

∥

∗



min

(

m,n

)

i=1

(

)

is the nuclear norm, σ

(

)

is the

ith largest singular value of P. Suppose that U



is the singular

value decomposition of P, we have U = (u

, . .., u

)∈ R

m×m

V =

(

, . . . , v

)

∈ R

n×n

. Note that U

(

, . . . , u

)

∈

r×m

and V

(

, . . . , v

)

∈ R

r×n

include truncated singular

values. The details about the truncated nuclear norm can be

referred to Hu et al. (2013). Clearly, an underlying projection

P can be learnt based on the problem to extract low-rank and

sparse features from data efficiently by embedding given data

into the learnt P, and the extracted features will hold the joint

low-rank and sparse properties. But note that the above problem

also obtains the underlying projection and adaptive weights from

original data, so the results may be inaccurate by the possibly

included noise. To handle this issue, we clearly borrow the idea

of rLatLRR (Zhang & Lin et al., 2014) by learning a projection

P based on the noise-removed clean data X-E instead of the

original data X to make the learnt features more notable and

robust to noise, where E is sparse error to be modeled. Note that

we also aim at computing the adaptive weights based on the

recovered clean data X-E so that the similarity measure can be

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