In this paper, we propose a speed-up approach for subclass discriminant analysis and formulate a novel efficient multi-view solution to it. The speed-up approach is developed based on graph embedding and spectral regression approaches that involve eigendecomposition of the corresponding Laplacian matrix and regression to its eigenvectors. We show that by exploiting the structure of the between-class Laplacian matrix, the eigendecomposition step can be substituted with a much faster process. Furthermore, we formulate a novel criterion for multi-view subclass discriminant analysis and show that an efficient solution for it can be obtained in a similar to the single-view manner. We evaluate the proposed methods on nine single-view and nine multi-view datasets and compare them with related existing approaches. Experimental results show that the proposed solutions achieve competitive performance, often outperforming the existing methods. At the same time, they significantly decrease the training time.
In this section, we focus on the spectral regression approach that was introduced as a way of speeding up the eigendecomposition step of LDA @cite_35 . It has been shown that the solution of the generalized eigendecomposition problem (10) is equivalent to the problem @math with the same eigenpairs, for @math and @math = @math : Exploiting this fact, the solution of (10) can be obtained by solving an eigenvalue decomposition problem @math and finding such @math that @math . In practice, such @math may not always exist, but it can be approximated with the closest value in the least squares sense: where @math is a regularization parameter and @math = @math .
@cite_35
1983622948 862919699 2085044339 2106253207
Linear discriminant analysis with nearest neighborhood classifier (LDA + NN) has been commonly used in face recognition, but it often confronts with two problems in real applications: (1) it cannot incrementally deal with the information of training instances; (2) it cannot achieve fast search against large scale gallery set. In this paper, we use incremental LDA (ILDA) and hashing based search method to deal with these two problems. Firstly two incremental LDA algorithms are proposed under spectral regression framework, namely exact incremental spectral regression discriminant analysis (EI-SRDA) and approximate incremental spectral regression discriminant analysis (AI-SRDA). Secondly we propose a similarity hashing algorithm of sub-linear complexity to achieve quick recognition from large gallery set. Experiments on FRGC and self-collected 100,000 faces database show the effective of our methods. Linear discriminant analysis (LDA) is a popular dimensionality reduction and classification method that simultaneously maximizes between-class scatter and minimizes within-class scatter. In this paper, we verify the equivalence of LDA and least squares (LS) with a set of dependent variable matrices. The equivalence is in the sense that the LDA solution matrix and the LS solution matrix have the same range. The resulting LS provides an intuitive interpretation in which its solution performs data clustering according to class labels. Further, the fact that LDA and LS have the same range allows us to design a two-stage algorithm that computes the LDA solution given by generalized eigenvalue decomposition (GEVD), much faster than computing the original GEVD. Experimental results demonstrate the equivalence of the LDA solution and the proposed LS solution. Linear discriminant analysis (LDA) has been a popular method for dimensionality reduction, which preserves class separability. The projection vectors are commonly obtained by maximizing the between-class covariance and simultaneously minimizing the within-class covariance. LDA can be performed either in the original input space or in the reproducing kernel Hilbert space (RKHS) into which data points are mapped, which leads to kernel discriminant analysis (KDA). When the data are highly nonlinear distributed, KDA can achieve better performance than LDA. However, computing the projective functions in KDA involves eigen-decomposition of kernel matrix, which is very expensive when a large number of training samples exist. In this paper, we present a new algorithm for kernel discriminant analysis, called Spectral Regression Kernel Discriminant Analysis (SRKDA). By using spectral graph analysis, SRKDA casts discriminant analysis into a regression framework, which facilitates both efficient computation and the use of regularization techniques. Specifically, SRKDA only needs to solve a set of regularized regression problems, and there is no eigenvector computation involved, which is a huge save of computational cost. The new formulation makes it very easy to develop incremental version of the algorithm, which can fully utilize the computational results of the existing training samples. Moreover, it is easy to produce sparse projections (Sparse KDA) with a L 1-norm regularizer. Extensive experiments on spoken letter, handwritten digit image and face image data demonstrate the effectiveness and efficiency of the proposed algorithm. Linear Discriminant Analysis (LDA) has been a popular method for extracting features that preserves class separability. The projection functions of LDA are commonly obtained by maximizing the between-class covariance and simultaneously minimizing the within-class covariance. It has been widely used in many fields of information processing, such as machine learning, data mining, information retrieval, and pattern recognition. However, the computation of LDA involves dense matrices eigendecomposition, which can be computationally expensive in both time and memory. Specifically, LDA has O(mnt + t3) time complexity and requires O(mn + mt + nt) memory, where m is the number of samples, n is the number of features, and t = min(m,n). When both m and n are large, it is infeasible to apply LDA. In this paper, we propose a novel algorithm for discriminant analysis, called Spectral Regression Discriminant Analysis (SRDA). By using spectral graph analysis, SRDA casts discriminant analysis into a regression framework that facilitates both efficient computation and the use of regularization techniques. Specifically, SRDA only needs to solve a set of regularized least squares problems, and there is no eigenvector computation involved, which is a huge save of both time and memory. Our theoretical analysis shows that SRDA can be computed with O(mn) time and O(ms) memory, where .s(les n) is the average number of nonzero features in each sample. Extensive experimental results on four real-world data sets demonstrate the effectiveness and efficiency of our algorithm.