We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized data points sampled with noise from a parameterized manifold, the local geometry of the manifold i...We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized data points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approximation for the tangent space at each point, and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases. We illustrate our algorithm using curves and surfaces both in 2D/3D Euclidean spaces and higher dimensional Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements.展开更多
Improved local tangent space alignment (ILTSA) is a recent nonlinear dimensionality reduction method which can efficiently recover the geometrical structure of sparse or non-uniformly distributed data manifold. In thi...Improved local tangent space alignment (ILTSA) is a recent nonlinear dimensionality reduction method which can efficiently recover the geometrical structure of sparse or non-uniformly distributed data manifold. In this paper, based on combination of modified maximum margin criterion and ILTSA, a novel feature extraction method named orthogonal discriminant improved local tangent space alignment (ODILTSA) is proposed. ODILTSA can preserve local geometry structure and maximize the margin between different classes simultaneously. Based on ODILTSA, a novel face recognition method which combines augmented complex wavelet features and original image features is developed. Experimental results on Yale, AR and PIE face databases demonstrate the effectiveness of ODILTSA and the feature fusion method.展开更多
Manifold learning has attracted considerable attention over the last decade,in which exploring the geometry and topology of the manifold is the central problem.Tangent space is a fundamental tool in discovering the ge...Manifold learning has attracted considerable attention over the last decade,in which exploring the geometry and topology of the manifold is the central problem.Tangent space is a fundamental tool in discovering the geometry of the manifold.In this paper,we will first review canonical manifold learning techniques and then discuss two fundamental problems in tangent space learning.One is how to estimate the tangent space from random samples,and the other is how to generalize tangent space to ambient space.Previous studies in tangent space learning have mainly focused on how to fit tangent space,and one has to solve a global equation for obtaining the tangent spaces.Unlike these approaches,we introduce a novel method,called persistent tangent space learning(PTSL),which estimates the tangent space at each local neighborhood while ensuring that the tangent spaces vary smoothly on the manifold.Tangent space can be viewed as a point on Grassmann manifold.Inspired from the statistics on Grassmann manifold,we use intrinsic sample total variance to measure the variation of estimated tangent spaces at a single point,and thus,the generalization problem can be solved by estimating the intrinsic sample mean on Grassmann manifold.We validate our methods by various experimental results both on synthetic and real data.展开更多
In this paper, the concepts of topological space and differential manifold are introduced, and it is proved that the surface determined by function F (x<sub>2</sub>, x<sub>2</sub>, …, x<sub...In this paper, the concepts of topological space and differential manifold are introduced, and it is proved that the surface determined by function F (x<sub>2</sub>, x<sub>2</sub>, …, x<sub>t</sub>) of class C<sup>r</sup> in Euelidean R<sup>t</sup> is a differential manifold. Using the intersection of the tangent plane and the hypernormal of the differential manifold to construct the shared master key of participants, an intuitive, secure and complete (t,n)-threshold secret sharing scheme is designed. The paper is proved to be safe, and the probability of successful attack of attackers is only 1/p<sup>t</sup><sup>-1</sup>. When the prime number p is sufficiently large, the probability is almost 0. The results show that this scheme has the characteristics of single-parameter representation of the master key in the geometric method, and is more practical and easy to implement than the Blakley threshold secret sharing scheme.展开更多
文摘We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized data points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approximation for the tangent space at each point, and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases. We illustrate our algorithm using curves and surfaces both in 2D/3D Euclidean spaces and higher dimensional Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements.
基金the National Natural Science Foundation of China(No.61004088)the Key Basic Research Foundation of Shanghai Municipal Science and Technology Commission(No.09JC1408000)
文摘Improved local tangent space alignment (ILTSA) is a recent nonlinear dimensionality reduction method which can efficiently recover the geometrical structure of sparse or non-uniformly distributed data manifold. In this paper, based on combination of modified maximum margin criterion and ILTSA, a novel feature extraction method named orthogonal discriminant improved local tangent space alignment (ODILTSA) is proposed. ODILTSA can preserve local geometry structure and maximize the margin between different classes simultaneously. Based on ODILTSA, a novel face recognition method which combines augmented complex wavelet features and original image features is developed. Experimental results on Yale, AR and PIE face databases demonstrate the effectiveness of ODILTSA and the feature fusion method.
基金supported by the National Natural Science Foundation of China(Grant No.60875044).
文摘Manifold learning has attracted considerable attention over the last decade,in which exploring the geometry and topology of the manifold is the central problem.Tangent space is a fundamental tool in discovering the geometry of the manifold.In this paper,we will first review canonical manifold learning techniques and then discuss two fundamental problems in tangent space learning.One is how to estimate the tangent space from random samples,and the other is how to generalize tangent space to ambient space.Previous studies in tangent space learning have mainly focused on how to fit tangent space,and one has to solve a global equation for obtaining the tangent spaces.Unlike these approaches,we introduce a novel method,called persistent tangent space learning(PTSL),which estimates the tangent space at each local neighborhood while ensuring that the tangent spaces vary smoothly on the manifold.Tangent space can be viewed as a point on Grassmann manifold.Inspired from the statistics on Grassmann manifold,we use intrinsic sample total variance to measure the variation of estimated tangent spaces at a single point,and thus,the generalization problem can be solved by estimating the intrinsic sample mean on Grassmann manifold.We validate our methods by various experimental results both on synthetic and real data.
文摘In this paper, the concepts of topological space and differential manifold are introduced, and it is proved that the surface determined by function F (x<sub>2</sub>, x<sub>2</sub>, …, x<sub>t</sub>) of class C<sup>r</sup> in Euelidean R<sup>t</sup> is a differential manifold. Using the intersection of the tangent plane and the hypernormal of the differential manifold to construct the shared master key of participants, an intuitive, secure and complete (t,n)-threshold secret sharing scheme is designed. The paper is proved to be safe, and the probability of successful attack of attackers is only 1/p<sup>t</sup><sup>-1</sup>. When the prime number p is sufficiently large, the probability is almost 0. The results show that this scheme has the characteristics of single-parameter representation of the master key in the geometric method, and is more practical and easy to implement than the Blakley threshold secret sharing scheme.
