Exponential Local Discriminant Embedding and Its Application to Face Recognition

Abstract

Local discriminant embedding (LDE) has been recently proposed to overcome some limitations of the global linear discriminant analysis method. In the case of a small training data set, however, LDE cannot directly be applied to high-dimensional data. This case is the so-called small-sample-size (SSS) problem. The classical solution to this problem was applying dimensionality reduction on the raw data (e.g., using principal component analysis). In this paper, we introduce a novel discriminant technique called "exponential LDE" (ELDE). The proposed ELDE can be seen as an extension of LDE framework in two directions. First, the proposed framework overcomes the SSS problem without discarding the discriminant information that was contained in the null space of the locality preserving scatter matrices associated with LDE. Second, the proposed ELDE is equivalent to transforming original data into a new space by distance diffusion mapping (similar to kernel-based nonlinear mapping), and then, LDE is applied in such a new space. As a result of diffusion mapping, the margin between samples belonging to different classes is enlarged, which is helpful in improving classification accuracy. The experiments are conducted on five public face databases: Yale, Extended Yale, PF01, Pose, Illumination, and Expression (PIE), and Facial Recognition Technology (FERET). The results show that the performances of the proposed ELDE are better than those of LDE and many state-of-the-art discriminant analysis techniques.

Variational quantum algorithms for dimensionality reduction and classification

“…The required gate resource count is O(log M K(K )). We next turn to find a matrix transform A that maximizes the local margins among different classes and pushes the homogenous samples closer to each other [66]. The overall process corresponds to the below mathematical formula:…”

“…After simple matrix algebra (seeing details in [66]), the columns of optimal A are the generalized eigenvectors with the l largest eigenvalues in…”

A Simple and Effective Generalization of Exponential Matrix Discriminant Analysis and Its Application to Face Recognition

“…The diffusion scale to the between-class distance is larger than that to the within-class distance. According to the inequality (11) and (13), the diffusion scale of GEDA is much larger than that of EDA. Hence, the distances between different class samples of GEDA are larger than that of EDA.…”

“…Then, the EDA method is introduced to solve this problem. The exponential LPP (ELPP) [9], the exponential DLPP method (EDLPP) [10], the exponential LDE method (ELDE) [11] and the exponential SDE method (ESDE) [12] are proposed. They are the exponential versions of the corresponding methods.…”

New algorithms for approximating $\varphi$-functions and their condition numbers for large sparse matrices

“…The ϕ-functions are defined for scalar arguments by the integral representation as follows This definition can be extended to matrices instead of scalars by using any of the available definitions of matrix functions [20,23]. In a wide range of applications, such as the matrix exponential discriminant analysis method for data dimensionality reduction [1,11,40,41,43,44], and the complex network analysis method based on matrix function [3,13,14,15], it is required to compute the matrix exponential with respect to large scale and low-rank matrix. In this paper, we are interested in computing several ϕ-functions consecutively, with respect to a large scale matrix A with low rank or with fast decaying singular values.…”