I Introduction
In the past a few years, inspired by advances in norm and norm techniques, sparse representation[4] has been widely applied in computer vision, such as image segmentation[25], image deblurring[30]
and face recognition
[24]. It is worth noting that Wright et al. proposed a sparse representation based classification(SRC) [24]to classify facial images, which is the first time to exploit the discriminative nature of sparse represeor face recognition. In fact, facial images have a high dimensionality, which usually lie on a lowdimensional subspace or submanifold. Thus, Yang
et al. [26]proposed a novel dimensionality reduction method that adopts SRC as a criterion to steer the design of a feature extraction method. In addition, many highdimensional data in real world may be better modeled by nonlinear manifolds. To overcome the nonlinear obstruction, some researches suggest to map these data into a kernel feature space by using some nonlinear mapping, and then SRC is performed in this new feature space by utilizing kernel trick
[31][28].However, most of the above work mainly focuses on the problem associated with vectorvalued data. The higherorder signals like images (2D, 3D or higher) have to be dealt with primarily by vectorizing them and applying any of the available vector techniques. As a result, such type vector features cannot efficiently characterize the highdimensional data in computer vision, machine learning and medical image analysis
[29]. Concretely, in traditional sparse representation based classification, the sparsity representation for each query image is attained by a dictionary composed of all gallery data across all classes in a linear combination way. Recent advance[19] suggests that encoding images through symmetric positive definite (SPD) matrices and then interpreting such matrices as points on Riemannian manifolds can lead to promising classification performance. For instance, the human facial images are regarded as samples from a nonlinear submanifold[22]. Unfortunately, the linear combination is not applicable to this case where data may be better modeled by nonlinear manifolds [10][8]. In other words, the direct applications of linear combination model to matrixvalued data will result in the comprised performance as inaccurate representation. Consequently, the traditional SRC is also no longer available to classification on SPD matrices as points on Riemannian manifolds.To address this problem, a few solutions have been recently proposed to generalize sparse coding problems to Riemannian manifolds, such as [3][9][17]. The most common approach is to calculate the tangent space to the manifold at the mean of the data points so as to obtain a Euclidean approximation of the manifold[19]. Inspired by this idea, Ho et al. [9] firstly proposed a nonlinear generalization of sparse coding to handle the nonlinearity of Riemannian manifolds, via flattening a SPD manifold using a fixed tangent space. In order to further measure the representation error effectively, in [17]
, a tensor sparse coding framework was proposed for positive definite matrices based on the logdeterminant divergence (Burg loss). Instead of using extrinsic similarity measures as work
[17], the authors[3] proposed to use the intrinsic Riemannian distance on the manifold of SPD matrices. Although locally flattening Riemannian manifolds via tangent spaces can handle their nonlinearity, it inevitably leads to very demanding computation due to switching back and forth between tangent spaces and the manifold[8]. Furthermore, linear reconstruction of SPD matrices is not as natural as in Euclidean space and this may incur errors [11]. On the other line, to address this nonlinear problems via LRR, a nonlinear LRR model is proposed to extend the traditional LRR from Euclidean space to Stiefel manifold [29], SPD manifold [5] and abstract Grassmann manifold [20] respectively. Lowrank representation based method, however, often suffers from high computational complexities as the nuclear norm regularized optimizing. From this view point, sparse representation based method can readily reduce the computational complexities greatly due to only solving norm optimization problems rather than nuclearnorm ones.The existing sparse representation methods on SPD matrices can been shown to be effective for classification[8][11], however, there still remain questions about classification in the multiple submanifolds setting[23] with sparse representation. As SPD matrices are often that lowdimensional data embedded in highdimensional nonEuclidean spaces, their underlying submanifolds are geodesic and referred to Riemannian multimanifolds. Let be a SPD matrix and hence a point on , it can be assumed residing on the tubular neighborhood of some unknown geodesic submanifold , of a Riemannian manifold. As for this issue, sparse representation based classification [8][11] has not been sufficiently explored yet. Another reason, may not trivial, the sparse coding coefficients may vary a lot even for similar query samples in classification task as the mechanism of minimization. As a result, the unsatisfied recognition rate will be achieved. Motivated by these observations, in this paper, we propose a neighborhood preserved sparse representation for robust classification on SPD matrices. When encoding the query sample, we aim to use the training samples lying in its vicinity as the training samples and the query sample may reside in the same submanifold leading to a better classification performance. Despite its simplicity, the proposed method performs well for classification task. Specifically, to thoroughly exploit the intrinsic geometry among data on Riemannian manifold, a neighborhood preserved prior induced from the geodesic distance, besides the sparsity, is imposed on the sparse coefficients so that the similar query data produce similar sparse codes.
The main contributions in our paper are summarized below.

To our best knowledge, it is the first attempts to formulate the local consistency into the sparse coding paradigm over a Riemannian manifold via embedding them into RKHS. It is significantly different from the work in [31][28] as the latter did not consider the Riemannian geometry structure within data.

To efficiently measure the neighborhood between data points on Riemannian manifold, we compare the two geodesic distance under Stein metric and LogEuclidean metric, respectively. To our best knowledge, this is one of the first attempts, from the weighted structured perspective, to compare the benefit from this two metrics for analyzing SPD matrices.

We apply our proposed methods to several classification tasks where the data are depicted as region covariance matrices.
The remainder of this paper is organized as follows. In Section II, we give a brief review on the related works. Section III is dedicated to introducing our novel neighborhood preserved kernel SRC, termed as NPKSRC. Section IV presents experimental results on image classification tasks. Finally, Section V concludes our paper and also provides the directions for future improvements.
Ii Related Work
Before we introduce our model, in this section, we briefly review the recent development of sparse representation based classification methods[24][9] and the analysis of Riemannian geometry of SPD manifold[14]. For convenience, Table II gives the notation used throughout this paper.
Notation  Description 
data matrix  
3D matrix or 3order tensor  
x  column vector 
the element at position of vector x  
the th entry of matrix  
norm of x  
norm of x  
transpose operator  
matrix Frobenius norm defined as  
nuclear norm of defined by the sum of its singular values 

matrix trace operator  
space of SPD matrices  
Log map  principal matrix logarithm 
Log map from SPD manifold to a tangent space at  
tangent space at a point on ,  
which is a vector space including the tangent vectors  
of all possible curves passing over . 
Iia Classification via Sparse Representation
Sparse representation based classification(SRC) has been wellknown as its robustness to face recognition[24]. Suppose that there exist classes and training data for each class . We denote by the collection of training data in the th class and by the collection of all training data over all classes. Given a test sample , which belongs to one of the classes, the goal of SRC is to find out the class to which belongs, by seeking its sparsest representation over all training data.
Concretely, the SRC solves the following optimization problem.
Once the problem is solved, the class of given test sample can be found as the class that best represents it using the corresponding training data in classwise way. That is, let be a vector whose only nonzero entries are the entries in that are associated with class , we can adopt the following rule to determine as class that has the minimum residual.
(1) 
IiB Riemannian Geometry on SPD Matrices
In general, SPD matrices lie on a nonflat Riemannian manifold, whose structure is suitably characterized by the geodesic distance induced by Riemannian metric. That is, a natural way to measure closeness of data on a Riemannian manifold is geodesics, eg. curves analogous to straight lines in . For any two data points on a manifold, geodesic distance is the length of the shortest curve on the manifold connecting them. For this reason, there are, currently, two popular distance measures in . One is the affine invariant Riemannian metric (AIRM) and the other is LogEuclidean metric.
As one of true metrics of geodesic distance, AIRM is probably the most widely used Riemannian metric defined as follows
[14]. Given , the AIRM of two tangent vectors is defined asThe geodesic distance between points induced from AIRM is then
(2) 
However, the above distance induced by AIRM is computationally intensive resulting in a significant numerical burden. To overcome this drawback of AIRM, LogEuclidean metric is defined on the Lie group of SPD matrices corresponding to a Euclidean metric in the logarithmic domain. Specifically, the distance under LogEuclidean metric is denoted by,
(3) 
IiC Spare Representation on SPD Matrices
Since SPD matrices belong to a Lie group which is a Riemannian manifold [1], it cripples many methods that rely on linear reconstruction. Generally, there are two methods to deal with the nonlinearity of Riemannian manifolds. One is to locally flatten the manifold to tangent spaces[19]. The underlying idea is to exploit the geometry of the manifold directly. The other is to map the data into a feature space usually a Hilbert space [10]. Precisely, it is to project the data into RKHS through kernel mapping [7]. Both of these methods are seeking a transformation so that the linearity reemerges.
A typical example of the former method is the one in [9]. Let be a SPD matrix and hence a point on . is a dictionary. An optimization problem for sparse coding of on a manifold is formulated as follows
(4) 
where denotes Log map from SPD manifold to a tangent space at , is the sparse vector and is the norm associated with . Because , the second term in Eq.(4) is essentially the error of linearly reconstructing by others on the tangent space of .
Although locally flattening Riemannian manifolds via tangent spaces[9] can handle their nonlinearity, it inevitably leads to very demanding computation due to switching back and forth between tangent spaces and the manifold. Furthermore, linear reconstruction of SPD matrices is not as natural as in Euclidean space and this may incur errors. Thus, the kernelbased sparse coding on SPD matrices is proposed as follows[6].
(5) 
where denotes a feature mapping function that projects SPD matrices into RKHS such that where is a positive definite (PD) kernel.
Iii Kernel Sparse Representation on SPD Matrices via Neighborhood Preserved
However, the constraint of norm sparsity is beneficial to classification task, a test input might be reconstructed by training images, i.e., codewords, which are far away from the test sample[21]. As a consequence, the SRC type methods will produce unsatisfying classification results. In addition, data of SPD matrices are often modeled as a union of lowdimensional submanifolds[23]. Under this context, classification algorithms aim at partitioning data based on the underlying lowdimensional nonEuclidean spaces. Therefore, the neighborhood of each data on Riemannian manifold can be fit by a geodesic submanifold model.
Motivated by the above issues, in this section, we propose a neighborhood preserved kernel sparse representation based classification (termed as NPKSRC) algorithm on SPD matrices, by considering the structure within data points. The formulation can be written as following.
(6) 
where means elementwise multiplication and is a vector imposing restriction on the structure of the solution. Similar to the prior work[6][9], the affine constraint is applied to our model too. Furthermore, by introducing structure constraint, i.e., , we actually enforce a smaller weight on the samples belonging to the same submanifold with the test input, and vice versa. To some extent, the entries of are denoting the affinity between the test input and the training data. Then, how to choose a informative entries of is a key factor of success for the subsequent classification tasks.
Iiia Analysis of the Weight Matrix
The structure of data are often determined by using pairwise distance between data points [21]. Moreover, manifold learning (neighborhood preservation model) has been widely used for dimension reduction by learning and embedding local consistency of original data into a lowdimensional representation[15][18]. For simplicity, we assume only there exist a twoclass data underlying geodesic submanifolds and , respectively. Given a test input , in general, there is a larger probability to assign it to that class determined by points lying on the submanifolds if the nearby points of is that points located on . From this intuition, we can use the affinity, i.e., geodesic distance, between test input and training data to compute the weight. Concretely, is constructed in terms of the geodesic distance of from every training sample(a subset of ). As such, a locally smooth sparse code vector is achieved where the sparsity is a result of the neighborhood preserving since the training samples far away from do not contribute to its reconstruction. Therefore, in this paper, we utilize the geodesic distance, under LogEuclidean metric, between a test input and training samples as the weight, illustrated as following.
IiiB The Proposed Method
Given the training data on SPD manifold, the corresponding kernel sparse representation algorithm is formulated as following.
(7) 
Through expanding the norm term and some algebra manipulations, we will consider the following problem that has a same solution to problem 7. For clarity and completeness, the detailed derivation of the problem 7 can be found in the appendix.
where and , given the SVD of is .
Here, we adopt LogEuclidean Gaussian kernel[11] to transform the SPD matrices into RKHS such that the linear combination will make sense. In contrast, the LogEuclidean kernel can well characterize the true geodesic distance between SPD matrices instead. Specifically, a LogEuclidean Gaussian kernel is defined by , which is a p.d. kernel for any .
IiiC Optimization
To solve the problem(7), we apply the wellknown alternating direction method of multipliers (ADMM)[2] here. Before directly using ADMM, we should decouple the variables in the problem (7) firstly. Let and introduce a variable . Then,
(8) 
where is a column vector whose entries are all ones.
The above problem is not convex in both, however, it is convex in a variable for fixed another unrelated one. Hence, the augmented Lagrangian function of problem (8) can be written as follows.
(9) 
Thus, we optimize the problem by alternatively fixing other unrelated variables as follows.

Update ,
(10) Setting the derivative w.r.t. to be zero gives the following.
where is the matrix of size with all ones.
Then,
(11) 
Update ,
(12) That is,
The above problem has the following closedform solution given by shrinkage operator[13]. That is,
(13) where is a shrinkage operator acting on each element of the given matrix, and is defined as .

Update and .
(14)
These iterative steps will be terminated when and are satisfied.
IiiD Classification
IiiE Complexity Analysis and Convergence
As for the computational cost of the proposed algorithm, it is mainly determined by the steps in ADMM. The total complexity of NPKSRC is, as a function of the number of data points, where is the total number of iterations. The soft thresholding to update the sparse matrix C in each step is relatively cheaper, much less than . For updating we can precompute the Cholesky decomposition of at the cost of less than , then compute new by using (11) which has a complexity of in general.
Iv Experimental Results
In this section, we present several experimental results to demonstrate the effectiveness of NPKSRC. To comprehensively evaluate the performance of NPKSRC, we tested it on texture images, human faces and pedestrian reidentification. Some sample images from test databases are shown in Figure 1.
We compare our proposed method with five stateoftheart methods in terms of recognition accuracy.

Sparse representation classification(SRC)[24] ;

Gabor featurebased sparse representation in Euclidean space (GSRC)[27] ;

Classification using Riemannian sparse representation based on Riemannian distance (RSRC)[3];

Classification using Riemnnian sparse representation based on Stein kernel (RSRS)[8];

LogEuclidean Gaussian kernel sparse representation based classification (LogEGkSRC) [11].
Iva Texture Classification
Firstly, we used Brodatz texture database to conduct classification task. In this dataset, it includes 5texture (‘5c’, ‘5m’, ‘5v’, ‘5v2’,‘5v3’), 10texture (‘10’, ‘10v’) and 16texture (‘16c’, ‘16v’) mosaics. Before using the proposed method, we downsampled each image into 256 256 and then split into 64 regions of size 32 32. To obtain their Region Covariance Matrices (RCM), a feature vector for any pixel is extracted, e.g., . Then, each region can be depicted by a 5 5 covariance descriptor. As for the obtained RCM, there are 64 covariance matrices in each class. We randomly selected 5 from each class as training samples and the rest as the query samples. That means almost 8% samples are selected as training data and the rest for testing classification. To achieve a stable result, the reported classification rate is averaged over 20 trials.
How to construct the weight vector is not a trivial work in our method. Here, to better characterize the local geometry within data, we test the two kinds of distance by different metrics, i.e., Stein metric and LogEuclidean metric. Brodatz‘16v’ is selected as test dataset for classification task, which includes 16 classes. The classification results are shown in Table II. As can be seen, the geodesic distance under LogEuclidean metric can better characterize the manifold structure of data by achieving a better classification rate.
Stein metric  LogE metric  
Accuracy  78.5  79.34 
To efficiently determine the parameters in our method, in Fig.2,we report the recognition accuracy on Brodatz‘16c’ with varying parameters and , respectively. From the figs., we can set and for the best recognition result.
By applying the different methods, we presented the classification results in Table III. As well, the tuned parameters are reported for the results achieved by other methods. The bold numbers highlight the best results. From the results, we can observe that the proposed approach outperforms other methods in most cases while, on average, RSRS achieved the second best performance based on Stein kernel. This can be interpreted by the Stein distance may better suit some subset of Brodatz dataset, i.e., ‘5v2’ and‘5v3’. As for SRC, it conducts classification as a baseline due to the lack of consideration of the intrinsic geometry structure within data.
‘5c’  ‘5m’  ‘5v’  ‘5v2’  ‘5v3’  ‘16v’  ‘16c’  ‘10v’  ‘10’  avg.  
SRC[24] (0.001)  20.00  20.00  20.00  20.00  20.00  6.25  44.07  10.00  20.00  20.04 
RSRC[3](0.01)  97.97  50.51  83.05  83.05  71.19  61.65  74.05  85.08  92.88  77.71 
RSRS[8](10.0)  98.31  89.15  83.05  87.12  88.14  72.46  83.79  88.31  94.24  87.17 
LogEGkSRC[11](0.001,0.02)  97.29  94.92  83.73  86.10  86.78  73.20  80.08  90.34  94.58  84.55 
NPKSRC(0.09,0.05)  98.31  98.98  84.41  84.07  87.12  79.34  88.35  91.02  98.31  89.26 
IvB Face Recognition
Next, we selected the “b” subset of FERET database to further evaluate the classification performance, in which it covers 1400 images with the size of 80 80 from 200 subjects (about 7 each). This subset consists of the images, under different expression and illumination conditions, marked by ‘ba’, ‘bd’, ‘be’, ‘bf’, ‘bg’, ‘bj’, and ‘bk’. Specifically, training images include neutral expression ‘ba’, smiling expression ‘bj’, and illumination changes ‘bk’, while test samples involve face images of varying pose angle such as ‘bd’+25, ‘be’+15, ‘bf’15, and ‘bg’25.
To represent a facial image, similar to the work [27], we created a 4343 region covariance matrix, i.e., a specific SPD matrix, which is composed of intensity value, spatial coordinates, 40 Gabor filters at 8 orientations and 5 scales. The downsampling factor in Gabor filtering is applied too. For SRC, the Gabor features are firstly vectorized and the common SRC classifier is applied. The classification results achieved by other methods are reported in Table IV and Fig.3. The tuned parameters are presented in the table too. For NPKSRC, the and are set 0.9 and 0.02, respectively. From the table, we can see our proposed method achieves pleasing recognition performance compared to others. This is owed to the consideration of locality structure between data by using Riemannian metric.
‘bg’  ‘bf’  ‘be’  ‘bd’  avg.  
SRC[24](0.01)  61.00  96.50  95.50  55.50  77.13 
GSRC[27](0.001)  79.00  97.00  93.50  77.00  86.60 
RSRC[3] (0.01)  79.50  97.00  96.50  68.50  85.38 
RSRS[8](10.0)  86.00  97.50  96.50  79.50  89.90 
NPKSRC (0.9, 0.02)  93.00  99.50  99.00  92.00  95.88 
IvC Pedestrian Reidentification
Finally, we conduct the person reidentification task by our proposed method and compare with other methods. Here, we used the modified ETHZ dataset[16], illustrated in Fig.4. The original ETHZ includes 3 Sequences, in which Sequence 1 contains 83 pedestrians (4,857 images), Sequence 2 contains 35 pedestrians (1,936 images), and Sequence 3 contains 28 pedestrians (1,762 images). To facilitate the subsequent processing, we first downsampled all images to 6432 pixels following the work[8]. To prepare the covariance descriptors, the following features are utilized: the position of pixel, the color information from RGB channels, the gradient and Laplacian information from the corresponding color part, respectively. That is, each region can be depicted by a 1717 covariance matrix. To constructing the training samples, 10 images are randomly selected from each subject while the rest are used for testing. For fairly comparison, we adopt five splits for each sequence to test the classification performance.
The recognition results are presented in TablesVVII. We tuned the parameters for each method to achieve the best results and reported them in tables. And the best results for each test Seq. are highlighted in bold numbers as usual. For the methods using kernel trick, the second parameter in the brackets denotes the kernel parameter. As can be seen, the proposed NPKSRC achieves the best score for each sequence in average sense. While for LogEGkSRC, it obtains the second best results in terms of classification rate thanks to the use of LogEuclidean metric. To explain this observation, it may owe to considering the weight structure within data. As for RSRS, it applies the Stein kernel inferior to the methods using LogEuclidean Gaussian one.
Furthermore, to clearly show the advantage of our method, we plot a recognition rate vs. each split for seq.1 in Fig.5. As the curves for seq.2 and seq.3 are similar to that of seq.1, we do not repeatedly present here.
avg.  
SRC[24](0.001)  87.97  87.06  88.36  89.39  90.04  88.56 
RSRC[3](0.01)  77.10  79.82  79.17  80.08  79.95  79.22 
RSRS[8](0.01,10)  89.78  89.39  91.72  92.63  91.98  91.10 
LogEGkSRC[11](0.001,0.02)  90.69  88.87  92.11  91.98  92.88  91.31 
NPKSRC (0.001,0.001)  92.11  90.30  91.98  92.88  92.76  92.00 
avg.  
SRC[24](0.001)  83.74  85.28  85.89  86.81  86.50  85.64 
RSRC[3](0.01)  84.66  81.60  82.52  86.20  83.44  83.68 
RSRS[8](0.01,10)  90.49  88.96  89.88  90.80  91.41  90.31 
LogEGkSRC[11](0.001,0.02)  91.41  89.88  88.65  92.94  90.80  90.74 
NPKSRC (0.001,0.001)  91.72  89.88  88.35  93.25  90.49  90.74 
avg.  
SRC[24](0.001)  95.92  95.10  90.61  93.88  93.06  93.71 
RSRC[3](0.01)  92.65  92.65  91.02  91.02  91.84  91.84 
RSRS[8](0.01,10)  98.78  97.55  98.37  96.73  98.37  97.96 
LogEGkSRC[11](0.001,0.02)  99.59  98.37  97.55  95.92  98.37  97.96 
NPKSRC (0.001,0.001)  99.18  99.59  97.14  97.14  98.37  98.28 
V Conclusion
In this paper, a novel robust classification algorithm, termed as neighborhood preserved sparse representation, is proposed for SPD matrices by fully exploiting the Riemannian geometry structure within data. Specifically, the local consistency constraint, formulated by the geodesic distance under LogEuclidean metric, is imposed onto the sparse coding paradigm over a Riemannian manifold. Experimental results show that the proposed method can provide better classification solutions than the stateoftheart approaches thanks to incorporating Riemannian geometry structure.
Although our proposed method achieved promising performance in terms of recognition rate, there still exists some open issues deserving to study. One prompt direction may be how to devise a better weighted constraint such that the beneficial discriminant representations can be achieved.
Vi Acknowledgement
The Project was supported in part by the Guangdong Natural Science Foundation under Grant (No. 2014A030313511) and in part by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, China.
Appendix
Given a leastsquares problem as following,
(15) 
where data and are on SPD manifold . This problem can be rewritten as a leastsquares problem on Euclidean space. That is,
(16) 
where and , given the SVD of is .
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