In recent years, a variety of nonlinear dimensionality reduction techniques (NLDR) have been proposed in the literature. They aim to address the limitations of traditional techniques such as PCA and classical scaling. Most of these techniques assume that the data of interest lie on an embedded non-linear manifold within the higher-dimensional space. They provide a mapping from the high-dimensional space to the low-dimensional embedding and may be viewed, in the context of machine learning, as a preliminary feature extraction step, after which pattern recognition algorithms are applied. Laplacian Eigenmaps (LE) is a nonlinear graph-based dimensionality reduction method. It has been successfully applied in many practical problems such as face recognition. However the construction of LE graph suffers, similarly to other graph-based DR techniques from the following issues: (1) the neighborhood graph is artificially defined in advance, and thus does not necessary benefit the desired DR task; (2) the graph is built using the nearest neighbor criterion which tends to work poorly due to the high-dimensionality of original space; and (3) its computation depends on two parameters whose values are generally uneasy to assign, the neighborhood size and the heat kernel parameter. To address the above-mentioned problems, for the particular case of the LPP method (a linear version of LE), L. Zhang et al.1 have developed a novel DR algorithm whose idea is to integrate graph construction with specific DR process into a unified framework. This algorithm results in an optimized graph rather than a predefined one.
Linear Dimensionality Reduction (LDR) techniques have been increasingly important in computer vision and
pattern recognition since they permit a relatively simple mapping of data onto a lower dimensional subspace,
leading to simple and computationally efficient classification strategies. Recently, many linear discriminant methods
have been developed in order to reduce the dimensionality of visual data and to enhance the discrimination
between different groups or classes. Many existing linear embedding techniques relied on the use of local margins
in order to get a good discrimination performance. However, dealing with outliers and within-class diversity
has not been addressed by margin-based embedding method. In this paper, we explored the use of different
margin-based linear embedding methods. More precisely, we propose to use the concepts of Median miss and
Median hit for building robust margin-based criteria. Based on such margins, we seek the projection directions
(linear embedding) such that the sum of local margins is maximized. Our proposed approach has been applied
to the problem of appearance-based face recognition. Experiments performed on four public face databases show
that the proposed approach can give better generalization performance than the classic Average Neighborhood
Margin Maximization (ANMM). Moreover, thanks to the use of robust margins, the proposed method down-grades
gracefully when label outliers contaminate the training data set. In particular, we show that the concept
of Median hit was crucial in order to get robust performance in the presence of outliers.
Linear Dimensionality Reduction (LDR) techniques have been increasingly important in computer vision and
pattern recognition since they permit a relatively simple mapping of data onto a lower dimensional subspace,
leading to simple and computationally efficient classification strategies. Recently, many linear discriminant methods
have been developed in order to reduce the dimensionality of visual data and to enhance the discrimination
between different groups or classes. Although many linear discriminant analysis methods have been proposed in
the literature, they suffer from at least one of the following shortcomings: i) they require the setting of many
parameters (e.g., the neighborhood sizes for homogeneous and heterogeneous samples), ii) they suffer from the
Small Sample Size problem that often occurs when dealing with visual data sets for which the number of samples
is less than the dimension of the sample, and iii) most of the traditional subspace learning methods have to
determine the dimension of the projected space by either cross-validation or exhaustive search. In this paper, we
propose a novel margin-based linear embedding method that exploits the nearest hit and the nearest miss samples
only. Our proposed method tackles all the above shortcomings. It finds the projection directions such that
the sum of local margins is maximized. Our proposed approach has been applied to the problem of appearancebased
face recognition. Experimental results performed on four public face databases show that the proposed
approach can give better generalization performance than the competing methods. These competing methods
used for performance comparison were: Principal Component Analysis (PCA), Locality Preserving Projections
(LPP), Average Neighborhood Margin Maximization (ANMM), and Maximally Collapsing Metric Learning algorithm
(MCML). The proposed approach could also be applied to other category of objects characterized by
large variations in their appearance.
Recently, a graph-based method was proposed for Linear Dimensionality Reduction (LDR). It is based on Locality
Preserving Projections (LPP). LPP is a typical linear graph-based dimensionality reduction (DR) method that
has been successfully applied in many practical problems such as face recognition. LPP is essentially a linearized
version of Laplacian Eigenmaps. When dealing with face recognition problems, LPP is preceded by a Principal
Component Analysis (PCA) step in order to avoid possible singularities. Both PCA and LPP are computed by
solving an eigen decomposition problem. In this paper, we propose a novel approach called "Selective Locality
Preserving Projections" that performs an eigenvector selection associated with LPP. Consequently, the problem
of dimension estimation for LPP is solved. Moreover, we propose a selective approach that performs eigenvector
selection for the case where the mapped samples are formed by concatenating the output of PCA and LPP. We
have tested our proposed approaches on several public face data sets. Experiments on ORL, UMIST, and YALE
Face Databases show significant performance improvements in recognition over the classical LPP. The proposed
approach lends itself nicely to many biometric applications.
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