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Tensor RPCA by Bayesian CP Factorization with Complex Noise

Qiong Luo

1,2

, Zhi Han

1∗

, Xi’ai Chen

1,2

, Yao Wang

, Deyu Meng

, Dong Liang

, Yandong Tang

State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences;

University of Chinese Academy of Sciences;

Xi’an Jiaotong University

{luoqiong, hanzhi, chenxiai, ytang}@sia.cn, {dymeng, liangdong}@mail.xjtu.edu.cn, yao.s.wang@gmail.com

Abstract

The RPCA model has achieved good performances in

various applications. However, two defects limit its effec-

tiveness. Firstly, it is designed for dealing with data in ma-

trix form, which fails to exploit the structure information of

higher order tensor data in some pratical situations. Sec-

ondly, it adopts L

-norm to tackle noise part which makes

it only valid for sparse noise. In this paper, we propose a

tensor RPCA model based on CP decomposition and model

data noise by Mixture of Gaussians (MoG). The use of ten-

sor structure to raw data allows us to make full use of the

inherent structure priors, and MoG is a general approxima-

tor to any blends of consecutive distributions, which makes

our approach capable of regaining the low dimensional lin-

ear subspace from a wide range of noises or their mixture.

The model is solved by a new proposed algorithm inferred

under a variational Bayesian framework. The superiority of

our approach over the existing state-of-the-art approaches

is demonstrated by extensive experiments on both of syn-

thetic and real data.

1. Introduction

In the ﬁelds of data analysis, principal component anal-

ysis (PCA) has been a classical and prevalent tool and has

extensive applications [16]. Originally, PCA aims to ﬁnd

the best L

-norm low-rank approximation of a speciﬁed

matrix due to its smoothness and has many fast numerical

solvers [9, 24, 25, 26, 35, 41]. But L

-norm is only suitable

for Gaussian noise and too susceptible to outliers and gross

noise. To increase the robustness of PCA, a series of works

have been conducted in recent years [12, 17, 13, 19].

Inspired by the improvement of low-rank matrix analy-

sis [4, 5, 30], the robust principal component analysis (RP-

CA) [40] has been proposed for remedying the deﬁciency of

traditional PCA, in which, a high dimensional observation

matrix is assumed to consist of a low-rank component and

∗

Corresponding author.

a sparse component. Speciﬁcally, let Y ∈ R

m×n

be the ob-

servation data matrix, X ∈ R

m×n

be the low-rank matrix,

E ∈ R

m×n

be the sparse noise matrix, and then we can

describe the RPCA as the following optimization problem:

min

X,E

X

∗

+ λE

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

where X

∗



(X) denotes the nuclear norm of

X, σ

(X) (r =1, 2, ..., min (m, n)) is the r

singular val-

ue of X, E



| denotes the L

-norm of E, and

is the element in the i

row and j

column of E. It has

been proved that if L and S satisfy a certain incoherence

condition, the RPCA can uniquely extract X and E from Y

[6]. RPCA has played an important role in handling vari-

ous problems, including robust matrix recovery [40], face

alignment [27], subspace segmentation [21] and so forth.

Recently, it has been noticed that more and more modern

applications contain data with a higher order tensor struc-

ture, such as background extraction [7], face recognition

and representation [40, 34, 38, 2], structure from motion

[36], object recognition [37] and motion segmentation [39].

Matrices can be viewed as second order tensors, howev-

er, moving from matrices to higher order tensors presents

signiﬁcant new challenges. A direct way to address these

challenges is to unfold tensors to matrices and then directly

apply the matrix RPCA model. Unfortunately, as recently

pointed out by [7], the multilinear structure is lost in such

matricization and as a result, methods constructed based on

these techniques often lead to suboptimal results. As such,

it is helpful to handle such raw data by using a direct ten-

sor representation, and several researches have been made

in the literatures [11, 20].

Moreover, L

-norm and L

-norm can characterize spe-

ciﬁc Laplace and Gaussian distributions, respectively, but

the real noise is generally not of a particular kind of noise

conﬁgurations, as already shown in [42]. Mixture of Gaus-

sians (MoG) is capable to commonly approximate wider

range of distributions due to its universal approximation ca-

pability, and Laplacian and Gaussian are regarded as a spe-

cial case of MoG [3]. It has been demonstrated that MoG

2017 IEEE International Conference on Computer Vision

DOI 10.1109/ICCV.2017.537

5029

(a) Observation High-

order Tensor

(b) Low-rank Tensor

Figure 1. TenRPCA-MoG sketch map. (a) is the observation ten-

sor, (b) is the low-rank tensor, (c) represents the complex noise.

can deal with complex noise in multiple computer vision

tasks, like image denoising and recovery [23, 42, 10].

In this paper, we propose a new tensor based RPCA

model with noise modeling by MoG, which is named as

TenRPCA-MoG. As shown in Fig. 1, the new model di-

vides the observation (noisy data) into a low-rank tensor

component (clean data) and the residue (noise), and models

them separately. It has the following contributions: ﬁrstly,

it treats raw high-order data as a tensor to reserve the com-

plete structure information, and uses CP tensor factorization

method to replace existing matrix factorization method to

extract low-rank structure in tensor data; secondly, it adopt-

s MoG to model noise which makes it have the ability to

ﬁt a wide range of noises rather than Gaussian or Lapla-

cian noise; thirdly, we formulate the problem as a genera-

tive model under the Bayesian framework and propose an

algorithm based on the variational inference theory to infer

the posterior and effectively solve the problem.

2. Related work

In order to make full use of high-order data structure in-

formation, Cao et al [7] extended the RPCA to the tensor

form based on Tucker decomposition and successfully ap-

plied it to the background extraction. Compared with other

methods, this model can achieve good extraction results at a

very low sampling rate. However, it is designed to deal with

only two types of noises, i.e., Gaussian noise and impulse

noise, such that it is inadequate for more complex noise in

real scenarios.

Meng and De la Torre [23] ﬁrstly applied the MoG

to low-rank matrix factorization (LRMF) for adapting to

unknown noise. Consequently, Zhao et al [42] proposed

a RPCA-MoG model which used MoG to model RPCA

noise. Beneﬁting from powerful approximation capability

of MoG, they successfully applied it to the face modeling

and background subtraction. However, these methods are

designed based on matrix techniques and fail to take the ad-

vantage of structure prior of original data. In order to over-

come such defect, Chen et al [10] developed a low-rank ten-

sor factorization (MoG-WLRTF) model with MoG and got

a good result on image denoising.

The differences and improvements of our model against

Chen [10] and Cao [7] are speciﬁed as follows: Chen’s work

is a LRTF model, while our work is a RPCA model under

the Bayesian framework, which has better adaptivity to var-

ious problems and the rank can be automatically conﬁrmed

by the algorithm itself; compared with our model, Cao’s

model lacks a universal noise modeling ability and is only

suitable for background extraction application.

3. Notations

Throughout the paper, lowercase letters (a, b, ···) de-

note scalars and bold lowercase letters denote vectors

(a, b, ···) with elements (a

, ···). Uppercase letter-

s (A, B, ···) denote the matrices with column vectors

, ···) and elements (a

, ···). High-order ten-

sors are represented by calligraphic letters (A, B, ···).A

K-mode tensor X∈R

×I

×···×I

is rank-one tensor

if it can be written as the out product of K vectors, i.e.,

X = a

◦ a

◦···◦a

4. TenRPCA-MoG model

The traditional RPCA model is formulated as Eq. 2. Ten-

sor RPCA has the similiar form as:

min

X ,E

X 

∗

+ λE

s.t. Y = X + E, (2)

where Y∈R

f×g×m

is the observation data tensor, X∈

f×g×m

is the low-rank tensor and E∈R

f×g×m

is the

noise tensor. Here, the L

norm is speciﬁcally set for deal-

ing with noise under sparse assumption, Laplacian noise,

for example. However, as we introduced before, the real

noise is generally much more complex rather than a simple

Laplacian noise. In order to improve the robustness to com-

plex noise of tensor RPCA, we introduce MoG for noise

modeling and obtain TenRPCA-MoG as:

min

X ,E

X 

∗

+ λE

s.t. Y = X + E, (3)

where symbol E

means that E is modeled with MoG.

As shown in Fig. 1 as a simulation, the ﬁrst and the second

Gaussians are for describing dense noise, while the third

Gaussian is for approximating Laplacian noise.

In our model, the low- rank factorization of tensor X is

obtained by CP decomposition. Hence, the detailed intro-

duction of our model starts from CP decomposition with its

helpful properties.

4.1. CP decomposition

There are two general tensor decomposition frameworks,

Tucker and CANDECOMP/PARAFAC (CP). CP decompo-

sition can be considered as a higher-order generalization of

the matrix singular value decomposition [8]. It decompos-

es a tensor into a sum of rank-one component tensors [18].

5030

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