Low-rank Decomposition And Its Applications In Computer Vision

In the era of information explosion,the emergence of massive data promotes the rapid development of machine learning,pattern recognition,computer vision and other areas in artificial intelligence,but also brings challenges to data processing at the same time.How to extract the intrinsic representation of the data and learn useful information from the data,is a key issue in these areas.Low-rank is an important property for describing the data structure and is suitable for extracting the essential characteristics of the data.This paper focuses on computational efficiency and robustness of the models,and studies the problem of low-rank matrix decomposition and its extensions to different applications from the perspective of utilizing label information and column space information,and preserving tensor structure information,followed by their applications in computer vision.The main tasks include:1.Fast and robust low-rank decomposition based on correntropy.Go Decomposition(GoDec)is an efficient low-rank matrix decomposition algorithm.However,optimal performance depends on sparse errors and Gaussian noise.The first problem to be solved in this paper is to decompose a matrix into low-rank component and unknown corruptions.By introducing a robust local similarity measure called correntropy to describe the corruptions and maximum correntropy criterion(MCC),based on GoDec,this paper proposes a more robust and faster lowrank decomposition algorithm: GoDec+.The proposed algorithm is solved by half-quadratic(HQ)optimization and greedy bilateral(GreB)paradigm.Based on the optimization theory on manifold,this paper analyses the convergence of GoDec+ rigorously.Experimental results show that GoDec+ is robust to different corruptions including the mixture of several noise(Gaussian noise,Laplacian noise,salt & pepper noise)and sparse corruption,and occlusion on both synthetic and real vision data,and is also fast.This paper further applies GoDec+ to more general applications including classification and subspace clustering.For classification,this paper constructs an ensemble subspace from the low-rank GoDec+ matrix and introduce an MCC-based classifier.For subspace clustering,this paper utilizes GoDec+'s low-rank matrix for MCC-based self-expression and combine it with spectral clustering.Face recognition,motion segmentation,and face clustering experiments show that the robustness of the proposed methods.2.Discriminative low-rank matrix decomposition.Although low-rank matrix decomposition methods have lots of applications in classification problem,most existing methods do not consider label information when learning low-rank structure.The second problem to be solved in this paper is how to incorporate label information in low-rank structure for better classification.The second work in this paper extends the proposed GoDec+ to discriminative GoDec+(D-GoDec+).In the proposed model,each class is represented by a shared underlying subspace and a specific transformation matrix.Structural label information and the Fisher discrimination criterion are incorporated to model the reconstruction errors and coefficients.An efficient solution to D-GoDec+ is proposed based on HQ optimization,and convergence of the solution is rigorously analyzed.Based on D-GoDec+,a simple yet effective classification method is presented by combining the discriminability of reconstruction errors and coefficients.Through the usage of transformation matrices,the classification method avoids complex encoding computation in the dictionary represented methods and thus is very efficient.Experimental results on face recognition,object classification,scene classification,and action recognition demonstrate the advantages of the proposed model.3.Bilinear low-rank matrix decompostion.Low-rank representation(LRR)has received intensive attention in graph-based subspace clustering and semi-supervised learning.However,most existing methods use original data as a dictionary directly without considering a sophisticated dictionary and focus more on row space while neglecting column space.The third problem to be solved in this paper is how to comprehensively utilize the information of both row space and column space for constructing a more representative dictionary.The third work in this paper extends GoDec+ to bilinear GoDec+(B-GoDec+).Benefiting from the form of bilinear representation,B-GoDec+ can learn the information of row space and column space simultaneously and unify structured dictionary learning and data representation in one model.An affinity matrix constructed from the learned data representation is employed in graph-based subspace clustering and semi-supervised learning.Theoretical analyses on the properties and convergence of B-GoDec+ are also given in this paper.Comprehensive experiments on popular machine learning research repositories in subspace clustering and semi-supervised learning demonstrate the superiority of the proposed method.4.Low-rank tensor decomposition.In the real world,a lot of data is in the form of tensor,while many low-rank decomposition methods take tensor as matrix or vector and thus destroy the tensor structure information.The forth problem to be solved in this paper is how to preserve tensor structure when applying low-rank decomposition.The forth work in this paper extends GoDec to tensor GoDec(T-GoDec)which handles data in the tensor case.It is difficult to find a low-n-rank approximation of a tensor since optimizing the rank of the matrix unfolding of one mode may change the one of another mode.By exploiting mode-d product and the data itself for self-representation,this paper pursues low-rank factor matrices to avoid pursuing a low-rank matrix of the matrix unfolding of each mode directly.The problem is solved by bilateral factorization and avoids time-consuming SVD.Benefiting from this representation,tensor GoDec can not only recover noisy data but also learn the affinity matrix of the data for subspace clustering.Experimental results show the efficiency and effectiveness of the proposed method.The algorithms proposed in this paper all have theoretical convergence analysis,and their effectiveness is also validated in the experiments on synthetic data and the applications of computer vision.This paper shows that low-rank is a good property for describing data structure and is flexible for applications in many practical problems.