1 Introduction
Manifold learning has drawn great interests since it was first proposed in 2000 ([1], [2], [4]
) as a promising nonlinear dimensionality reduction (NDR) method for highdimensional data manifolds. Its basic assumption is that highdimensional input data samples lie on or close to a lowdimensional smooth manifold embedded in the ambient Euclidean space. For example, by rotating the camera around the same object with fixed radius, images of the object can be viewed as a onedimensional curve embedded in a highdimensional Euclidean space, whose dimension equals to the number of pixels in the image. With the manifold assumption, manifold learning methods aim to extract the intrinsic degrees of freedom underlying the input highdimensional data samples, by preserving local or global geometric characteristics of the manifold from which data samples are drawn. In recent years, various manifold learning algorithms have been proposed, such as locally linear embedding (LLE)
[2, 3], ISOMAP [4, 5], Laplacian eigenmap (LE) [12], diffusion maps (DM) [14], local tangent space alignment (LTSA) [11], and Riemannian manifold learning [13]. They have achieved great success in finding meaningful lowdimensional embeddings for highdimensional data manifolds. Meanwhile, manifold learning also has many important applications in realworld problems, such as human motion detection [17], human face recognition
[18], classification and compressed expression of hyperspectral imageries [19], dynamic shape and appearance classification [20], and visual tracking [21, 22, 23].However, a main drawback of the manifold learning methods is that they learn the lowdimensional representations of the highdimensional input data samples implicitly. No explicit mapping relationship from the input data manifold to the output embedding can be obtained after the training process. Therefore, in order to obtain the lowdimensional representations of the new coming samples, the learning procedure, containing all previous samples and new samples as inputs, has to be repeatedly implemented. It is obvious that such a strategy is extremely timeconsuming for sequentially arrived data, which greatly limits the application of the manifold learning methods to many practical problems, such as classification, target detection, visual tracking and detection.
In order to address the issue of lacking explicit mappings, many linear projection based methods have been proposed for manifold learning by assuming that there exists a linear projection between the highdimensional input data samples and their lowdimensional representations, such as Locality Preserving Projections (LPP) [24, 25], Neighborhood Preserving Embedding (NPE) [26], Neighborhood Preserving Projections (NPP) [27], Orthogonal Locality Preserving Projections (OLPP) [28], Orthogonal Neighborhood Preserving Projections (ONPP) [29, 30], and Graph Embedding [31]. Although these methods have achieved their success in many problems, the linearity assumption may still be too restrictive.
On the other hand, several kernelbased methods have also been proposed to give nonlinear but implicit mappings for manifold learning (see, e.g. [32, 33, 34, 35]). These methods reformulate the manifold learning methods as kernel learning problems and then utilize the existing kernel extrapolation techniques to find the location of new data samples in the lowdimensional space. The mappings provided by the kernelbased methods are nonlinear and implicit. Furthermore, the performance of these methods depends on the choice of the kernel functions, and their computational complexity is extremely high for very large data sets.
In this paper, an explicit nonlinear mapping for manifold learning is proposed for the first time, based on the assumption that there exists a polynomial mapping from the highdimensional input data samples to their lowdimensional representations. The proposed mapping has the following main features.

The mapping is explicit, so it is straightforward to locate any new data samples in the lowdimensional space. This is different from the traditional manifold learning methods such as like LLE, LE, and ISOMAP [4] in which the mapping is implicit and it is not clear how new data samples can be embedded in the lowdimensional space. Compared with kernelbased mappings, the proposed mapping does not depend on the specific kernels in finding the lowdimensional representations of new data samples.

The mapping is nonlinear. In contrast to the linear projectionbased methods which find a linear projection mapping from the input highdimensional samples to their lowdimensional representations, the proposed mapping provides a nonlinear polynomial mapping from the input space to the reduced space. Clearly, it is more reasonable to use a polynomial mapping to handle with data samples lying on nonlinear manifolds. Meanwhile, our analysis and experiments show that the proposed mapping is of similar computational complexity with the linear projectionbased methods.
Combining this explicit nonlinear mapping with existing manifold learning methods (e.g. LLE, LE, Isomap) can give explicit manifold learning algorithms. In this paper, we concentrate on the LLE manifold learning method and propose an explicit nonlinear manifold learning algorithm called Neighborhood Preserving Polynomial Embedding (NPPE) algorithm. Experiments on both synthetic and realworld data have been conducted to illustrate the validity and effectiveness of the proposed mapping.
The remaining part of the paper is organized as follows. Section 2 gives a brief review of the existing manifold learning methods including those based on linear projections and kernelbased nonlinear mappings. Details of the explicit nonlinear mapping for manifold learning are presented in Section 3, whilst the NPPE algorithm is given in Section 4. In Section 5, experiments are conducted on both synthetic and realworld data sets to demonstrate the validity of the proposed algorithm. Conclusion is given in Section 6.
2 Related Works
In this section, we briefly review existing manifold learning algorithms including those based on linear projections and outofsample nonlinear extensions for learned manifolds.
For convenience of presentation, the main notations used in this paper are summarized in Table I
. Throughout this paper, all data samples are in the form of column vectors. Matrices are expressed using normal capital letters and data vectors are represented using lowercase letters. The superscript of a data vector is the index of its component.
dimensional Euclidean space where input samples lie  
dimensional Euclidean space, , where the  
lowdimensional embedding lie  
, the th input sample in ,  
, the set of input samples  
, matrix of input samples  
, lowdimensional representation  
of obtained by manifold learning,  
, the set of lowdimensional  
representations  
, matrix of lowdimensional  
representations  
Identity matrix of size  
norm where for an  
dimensional vector 
2.1 Manifold Learning Methods
According to the geometric characteristics which are preserved, existing manifold learning methods can be cast into two categories: local or global approaches.
As local approaches, Locally Linear Embedding (LLE) [2, 3] preserves local reconstruction weights. Locally Multidimensional Scaling (LMDS) [9]
preserves local pairwise Euclidean distances among data samples. Maximum Variance Unfolding (MVU)
[10] also preserves pairwise Euclidean distances in each local neighborhood, but it maximizes the variance of the lowdimensional representations at the same time. Local Tangent Space Alignment (LTSA) [11] keeps the local tangent structure. Diffusion Maps [14] preserves local pairwise diffusion distances from highdimensional data to the lowdimensional representations. Laplacian Eigenmap (LE) [12] preserves the local adjacency relationship.As global approaches, Isometric Feature Mapping (ISOMAP) [4, 5] preserves the pairwise geodesic distances among the highdimensional data samples and their lowdimensional representations. Hessian Eigenmaps (HLLE) [15] extends ISOMAP to more general cases where the set of intrinsic degrees of freedom may be nonconvex. In Riemannian Manifold Learning (RML) [13], the coordinates of data samples in the tangential space are preserved to be their lowdimensional representations.
2.2 Linear Projections for Manifold Learning
Manifold learning algorithms based on linear projections assume that there exists a linear projection which maps the highdimensional samples into a lowdimensional space, that is,
(1) 
where is a highdimensional sample and is its lowdimensional representation. Denote by the th column of . Then from a geometric point of view, data samples in are projected into an dimensional linear subspace spanned by . The lowdimensional representation is the coordinate of in with respect to the basis .
2.2.1 Lpp
Locality Preserving Projections (LPP) [24, 25] provides a linear mapping for Laplacian Eigenmaps (LE), by applying (1) into the training procedure of LE. The LE method aims to train a set of lowdimensional representations which can best preserve the adjacency relationship among highdimensional inputs . If and are “close” to each other, then and should also be so. This property is achieved by solving the following constrained optimization problem
(2)  
(3) 
where the penalty weights are given by the heat kernel and .
In LPP, equation (1) is applied to (2) and (3), that is, each is replaced with . By a straightforward algebraic calculation, equations (2) and (3) are transformed into
(4)  
(5) 
where , and is the diagonal matrix whose th entry is
. This optimization problem leads to a generalized eigenvalue problem
and the optimal solutions
are the eigenvectors corresponding to the
smallest eigenvalues.Once are computed, the linear projection matrix provided by LPP is given by . For any new data sample from the highdimensional space , LPP finds its lowdimensional representation simply by .
2.2.2 NPP and NPE
The linear projection mapping for Locally Linear Embedding (LLE) is independently provided by Neighborhood Preserving Embedding (NPE) [26] and Neighborhood Preserving Projections (NPP) [27]. Similarly to LPP, NPE and NPP apply the linear projection assumption (1) to the training process of LLE and reformulate the optimization problem in LLE as to compute the linear projection matrix.
During the training procedure of LLE, a set of linear reconstruction weights are first computed by solving a convex optimization problem
where is the index set of the nearest neighbors of . Then LLE aims to preserve from to . This is achieved by solving the following optimization problem
(6)  
(7) 
In NPE and NPP, the linear projection assumption (1) is used in the above optimization problem, so (6) and (7) become
(8)  
(9) 
where with . The optimal solutions are the eigenvectors of the following generalized eigenvalue problem corresponding to the smallest eigenvalues
After finding the linear projection matrix , any new data sample from the highdimensional space can be easily mapped into the lower dimensional space by .
2.2.3 OLPP and ONPP
Orthogonal Locality Preserving Projections (OLPP) [28] and Orthogonal Neighborhood Preserving Projections (ONPP) [29, 30] are the same as LPP and NPE (or NPP), respectively, except that the linear projection matrix provided by LPP and NPE (or NPP) is restricted to be orthogonal. This is achieved by replacing the constraints (5) and (9) with . Then the optimization problems in OLPP and ONPP become
OLPP:  (10)  
ONPP:  (11) 
Unlike in the cases of LPP and NPE (or NPP), these two optimization problems lead to eigenvalue problems which are much easier to solve numerically than a generalized eigenvalue problem. The column vectors of are given by the eigenvectors of corresponding to the smallest eigenvalues. The same result holds for by replacing with . The reader is referred to [28] and [29, 30] for details of these two algorithms.
2.3 OutofSample Nonlinear Extensions for Manifold Learning
Besides linear projections for manifold learning, several outofsample nonlinear extensions are also proposed for manifold learning in order to get lowdimensional representations of unseen data samples from the learned manifold. These methods are based on kernel functions and extrapolation techniques. A common strategy taken by these methods is to reformulate manifold learning methods as kernel learning problems. Then extrapolation techniques are employed to find the location of new coming samples in the lowdimensional space from the learned manifold. Bengio et al. [32, 36]
proposed a unified framework for extending LLE, ISOMAP and LE, in which these methods are seen as learning eigenfunctions of operators defined from datadependent kernels. The datadependent kernels are implicitly defined by LLE, ISOMAP LE and are used together with the Nyström formula
[38] to extrapolate the embedding of a manifold learned from finite training samples to new coming samples for LLE, ISOMAP and LE (see [32, 36]). Chin and Suter [35]investigated the equivalence between MVU and Kernel Principal Component Analysis (KPCA)
[39], by which extending MVU to new samples is reduced to extending a kernel matrix. In their work [35], the kernel matrix is generated from an unknown kernel eigenfunction which is approximated using Gaussian basis functions. A framework was proposed in [33] for efficient kernel extrapolation which is based on a matrix approximation theorem and an extension of the representer theorem. Under this framework, LLE was reformulated and the issue of extending LLE to new data samples was addressed in [33].3 Explicit Nonlinear Mappings for Manifold Learning
In this section, we propose an explicit nonlinear mapping for manifold learning, based on the assumption that there is a polynomial mapping between the highdimensional data samples and their lower dimensional representations. Precisely, given input samples and their low dimensional representations , we assume that there exists a polynomial mapping which maps to , that is, the th component of is a polynomial of degree with respect to in the following manner:
(12) 
where are all integers. The superscript stands for the tuple indexing array and is the vector of polynomial coefficients which is defined by
(13) 
By assuming the polynomial mapping relationship, we aim to find a polynomial approximation to the unknown mapping from the highdimensional data samples into their lowdimensional embedding space. Compared with the linear projection assumption used previously, a polynomial mapping provides highorder approximation to the unknown nonlinear mapping and therefore is more accurate for data samples lying on nonlinear manifolds.
In order to apply this explicit nonlinear mapping to manifold learning algorithms, we need two definitions from matrix analysis [40].
Definition 3.1
The Kronecker product of an matrix and a matrix is defined as
which is an matrix.
Definition 3.2
The Hadamard product of two matrices and is defined as
Recently, it was proved in [31] that most manifold learning methods, including LLE, LE, and ISOMAP, can be cast into the framework of spectral embedding. Under this framework, finding the lowdimensional embedding representations of the highdimensional data samples is reduced to solving the following optimization problem
(14)  
(15) 
where , , are positive weights which can be defined by using the input data samples and .
Applying the polynomial assumption (12) to the above general model of manifold learning gives a general manifold learning algorithm with an explicit nonlinear mapping. Denote by and substitute (12) into (14). Then the objective function becomes
(16)  
Substitute (12) into (15), so the constraint is transformed into
This is equivalent to
(17) 
where for and otherwise.
In order to simplify (16) and (17), we define by
(18) 
Then , so (16) and (17) are reduced, respectively, to
(19)  
(20) 
By writing , (19) and (20) can be further simplified to
(21)  
(22) 
where and is a diagonal matrix whose th diagonal entry is .
By the RayleighRitz Theorem [40], the optimal solutions are the eigenvectors of the following generalized eigenvalue problem corresponding to the smallest eigenvalues
(23) 
Once are computed, the explicit nonlinear mapping from the highdimensional data samples to the lowdimensional embedding space can be given as
(24) 
where is a highdimensional data sample and is its lowdimensional representation. For a new coming sample , its location in the lowdimensional embedding manifold can be simply obtained by
(25) 
where is defined in the same way as in (18).
In the next section, we will make use of a similar method as in LLE to define the weights , so that the geometry of the neighborhood of each data point can be captured.
4 Neighborhood Preserving Polynomial Embedding
In this section, we propose a new manifold learning algorithm with an explicit nonlinear mapping, named Neighborhood Preserving Polynomial Embedding (NPPE), which is obtained by defining the weights , in a way similar to the LLE method and combining them with the explicit nonlinear mapping as in the preceding Section 3.
4.1 Nppe
Consider a data set from the highdimensional space NPPE starts with finding a set of linear reconstruction weights which can best reconstruct each data point by its nearest neighbors (kNNs). This step is identical with that of LLE [2, 3]. The weights , which are defined to be nonzero only if is among the NNs of , are computed by solving the following optimization problem
(26) 
The weights represent the linear coefficients for reconstructing the sample from its neighbors , whilst the constraint means that is approximated by a convex combination of its neighbors. The weight matrix, , has a closedform solution given by
(27) 
where is a column vector formed by the nonzero entries in the th row of and is a column vector of all ones. The th entry of the matrix is , where and are among the NNs of .
NPPE aims to preserve the reconstruction weights from the highdimensional input data samples to their lowdimensional representations under the polynomial mapping assumption. This is achieved by solving the following optimization problem
(28) 
where each satisfies (12).
By a simple algebraic calculation, it can be shown that (28) is equivalent to (14) and (15) with
(29) 
By the result in Section 3, the explicit nonlinear mapping can be obtained by solving (23) and the lowdimensional representations of can be computed by applying (24) to . For a new coming sample , its lowdimensional representation can be simply given by (25).
We conclude this section by summarizing the NPPE algorithm in Algorithm 1.
4.2 Computational Complexity and Simplified NPPE
In the training procedure of NPPE, the computational complexity of generating is . Computing and takes and operations, respectively, since there are only nonzero entries in each column of and is a diagonal matrix. The computational complexity of the final eigendecomposition is , which is the most timeconsuming step.
In the procedure of locating new samples with NPPE, generating takes operations and computing takes operations.
From the above analysis, it can be seen that, as the polynomial order increases, the overall computational complexity increases exponentially with , which would be extremely timeconsuming when the data dimension is very high. To address this issue, we simplify NPPE by removing the crosswise items. This is achieved by replacing the Kronecker product in (18) with the Hadamard product
(30) 
With this strategy, the computational complexity of generating is reduced to , whilst the computational complexity computing is reduced to . The Simplified NPPE (SNPPE) is summarized in Algorithm 2.
Finally, the computational complexity of SNPPE, linear methods and kernel methods on computing is summarized in Table II. The computational complexity of different kernel methods varies. Here we only state the computational complexity of the common step of computing the inner products. It is obvious that the total complexity in computing is not less than this value.
Methods  SNPPE  Linear  Kernel 

Complexity 
4.3 Discussion
In this subsection, we briefly explain why NPPE or SNPPE has a better performance than its linear counterparts for nonlinearly distributed data sets.
Let be a nonlinear map from a manifold to such that , where is at least thorder differentiable. For simplicity, and without loss of generality we may assume that and that . Then the Taylor expansion of at zero is given by
(31) 
where and are the gradient and Hessian of , respectively. From (31), it can be seen that the linear methods only use the firstorder approximation provided by to approximate the nonlinear mapping , while the proposed polynomial mapping contains the extra highorder terms. Therefore, the explicit nonlinear mapping based on the polynomial assumption gives a better approximation to the true nonlinear mapping than the explicit linear one.
5 Experimental Tests
In this section, experiments on both synthetic and real world data sets are conducted to illustrate the validity and effectiveness of the proposed NPPE algorithm. In Section 5.1, NPPE is tested on recovering geometric structures of surfaces embedded in . In Section 5.2, NPPE is applied to locating new coming data samples in the learned lowdimensional space. In Section 5.3, NPPE is used to extract intrinsic degrees of freedom underlying two image manifolds. In the experiments, the simplified version of NPPE is implemented and compared with NPP [27] and ONPP [30] (which apply the linear and orthogonal linear projection mapping to the training procedure for LLE, respectively) as well as the kernel extrapolation (KE) method proposed in [33].
There are two parameters in the NPPE algorithm, the number of nearest neighbors and the polynomial degree . is usually set to be of the number of training samples, and the experimental tests show that NPPE is stable around this number. The choice of depends on the dimension . When is small, can be large to make NPPE more accurate. When is large, should be small to make NPPE computationally efficient. Experiments show that NPPE with is already accurate enough.
5.1 Learning Surfaces in with NPPE
In the first experiment, NPPE, NPP, ONPP and LLE are applied to the task of unfolding surfaces embedded in . The surfaces are the SwissRoll, SwissHole, and Gaussian, all of which are generated by the Matlab Demo available at http://www.math.umn.edu/~wittman/mani/. On each manifold, data samples are randomly generated for training. The number of nearest neighbors is and the polynomial degree . The experimental results are shown in Fig. 1. In each subfigure, stands for the generating data such that , where is the nonlinear mapping that embeds in . It can be seen from Fig. 1 that NPPE outperforms all the other three methods, even the LLE method itself. NPP and ONPP fail to unfold these nonlinear manifolds (except for ONPP on Gaussian).
Furthermore, in order to estimate the similarity between the learned lowdimensional representations and the generating data, the residual variance
[4] is computed, where is the standard linear correlation coefficient taken over all entries of and . The lower is, the more similar and are. The estimation results are shown in Fig. 1(d). It can be seen that the embedding given by NPPE is the most similar one.5.2 Locating New Data Samples with NPPE
In the second experiment, we apply NPPE, NPP, ONPP and KE to locating new coming samples in the learned lowdimensional space. First, data samples which evenly distribute on the SwissRoll manifold are generated. Then samples are randomly selected as the training data to learn the mapping relationship from to by NPPE, NPP, ONPP and KE. The learned mappings are used to provide the lowdimensional representations for the rest samples. The time cost of computing the lowdimensional representations of the testing samples is also recorded. Experimental results are shown in Fig. 2. It can be seen that NPPE not only gives the best locating result but also has much lower time cost than KE. NPP and ONPP are faster for computation but fail to give the correct embedding result. The same experiment is also conducted on data samples randomly selected from SwissRoll. The results are shown in Fig. 3. NPPE also outperforms the other three methods.
To further validate the performance of NPPE, we randomly generate samples on the SwissRoll manifold, for training and for testing. The experimental procedure is just the same as the preceding one. Time cost versus number of testing samples is shown in Fig. 4(a). The residual variances between the generating data of the testing samples and their lowdimensional representations given by the four methods, are illustrated in Fig. 4(b). The experimental results show that NPPE is more accurate than all the other three methods with a similar computational cost with NPP and ONPP. Note that, in all the above experiments, the time cost of KE is increasing linearly with the number of testing samples increasing, whilst that of NPP, ONPP and NPPE is almost the same with the increase of the number of testing samples.
5.3 Learning Image Manifolds with NPPE
In the last experiment, NPPE is applied to extract intrinsic degrees of freedom underlying two image manifolds, the lleface [2] and usps0.
The lleface consists of face images of the same person at resolution , and the two intrinsic degrees of freedom underlying the face images are rotation of the head and facial emotion. We randomly select samples as the training data and samples as the testing data. The number of nearest neighbors is set to be . The experimental results are shown in Fig. 5. The training and testing results are shown on the left and right columns, respectively, in Fig. 5. training samples and testing samples are randomly selected and attached to the learned embedding. It can be seen that NPPE and NPP have successfully recovered the underlying structure of lleface, while the result given by KE is not satisfactory. The rotation degree is not extracted by the learned embedding with KE. Time cost on locating new data samples by these three methods is shown in Fig. 7(a). The time cost of NPPE is higher than that of NPP but lower than that of KE, which supports the analysis of computational complexity in Section 4.2.
The usps0 data set consists of images of handwritten digit ‘0’ at resolution , and the two underlying intrinsic degrees of freedom are the line width and the shape of ‘0’. samples are randomly selected as training data and samples are chosen to be testing data. The number of nearest neighbors is set to be . Fig. 6 illustrates the experimental results. Training and testing results are shown on the left and right columns, respectively. training samples and testing samples are randomly selected and shown in the learned embedding. It can be seen that NPPE has successfully recovered the underlying structure, while it is hard to see the changes of line width and shape in the embedding given by KE and ONPP. Time cost on locating new data samples by these three methods is shown in Fig. 7(b). The time cost of NPPE is higher than ONPP but much lower than KE.
6 Conclusion
In this paper, an explicit nonlinear mapping for manifold learning is proposed for the first time. Based on the assumption that there is a polynomial mapping from the highdimensional input samples to their lowdimensional representations, an explicit polynomial mapping is obtained by applying this assumption to a generic model of manifold learning. Furthermore, the NPPE algorithm is a nonlinear dimensionality reduction technique with a explicit nonlinear mapping, which tends to preserve not only the locality but also the nonlinear geometry of the highdimensional data samples. NPPE can provide convincing embedding results and locate new coming data samples in the reduced lowdimensional space simply and quickly at the same time. Experimental tests on both synthetic and realworld data have validated the effectiveness of the proposed NPPE algorithm.
Experiment on locating new samples for uniformly distributed
SwissRoll data. (a) Training data and their generating data. (b) Time cost versus number of testing samples. (c) Locating results by NPPE. (d) Locating results by NPP. (e) Locating results by ONPP. (f) Locating results by KE. In (c)(f), stands for the training result.References
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