Manifold Elastic Net: A Unified Framework for Sparse Dimension Reduction

It is difficult to find the optimal sparse solution of a manifold learning based dimensionality reduction algorithm. The lasso or the elastic net penalized manifold learning based dimensionality reduction is not directly a lasso penalized least square problem and thus the least angle regression (LARS) (Efron et al. \cite{LARS}), one of the most popular algorithms in sparse learning, cannot be applied. Therefore, most current approaches take indirect ways or have strict settings, which can be inconvenient for applications. In this paper, we proposed the manifold elastic net or MEN for short. MEN incorporates the merits of both the manifold learning based dimensionality reduction and the sparse learning based dimensionality reduction. By using a series of equivalent transformations, we show MEN is equivalent to the lasso penalized least square problem and thus LARS is adopted to obtain the optimal sparse solution of MEN. In particular, MEN has the following advantages for subsequent classification: 1) the local geometry of samples is well preserved for low dimensional data representation, 2) both the margin maximization and the classification error minimization are considered for sparse projection calculation, 3) the projection matrix of MEN improves the parsimony in computation, 4) the elastic net penalty reduces the over-fitting problem, and 5) the projection matrix of MEN can be interpreted psychologically and physiologically. Experimental evidence on face recognition over various popular datasets suggests that MEN is superior to top level dimensionality reduction algorithms.Comment: 33 pages, 12 figure

2D Face Recognition System Based on Selected Gabor Filters and Linear Discriminant Analysis LDA

We present a new approach for face recognition system. The method is based on 2D face image features using subset of non-correlated and Orthogonal Gabor Filters instead of using the whole Gabor Filter Bank, then compressing the output feature vector using Linear Discriminant Analysis (LDA). The face image has been enhanced using multi stage image processing technique to normalize it and compensate for illumination variation. Experimental results show that the proposed system is effective for both dimension reduction and good recognition performance when compared to the complete Gabor filter bank. The system has been tested using CASIA, ORL and Cropped YaleB 2D face images Databases and achieved average recognition rate of 98.9 %

Penalized Orthogonal Iteration for Sparse Estimation of Generalized Eigenvalue Problem

We propose a new algorithm for sparse estimation of eigenvectors in generalized eigenvalue problems (GEP). The GEP arises in a number of modern data-analytic situations and statistical methods, including principal component analysis (PCA), multiclass linear discriminant analysis (LDA), canonical correlation analysis (CCA), sufficient dimension reduction (SDR) and invariant co-ordinate selection. We propose to modify the standard generalized orthogonal iteration with a sparsity-inducing penalty for the eigenvectors. To achieve this goal, we generalize the equation-solving step of orthogonal iteration to a penalized convex optimization problem. The resulting algorithm, called penalized orthogonal iteration, provides accurate estimation of the true eigenspace, when it is sparse. Also proposed is a computationally more efficient alternative, which works well for PCA and LDA problems. Numerical studies reveal that the proposed algorithms are competitive, and that our tuning procedure works well. We demonstrate applications of the proposed algorithm to obtain sparse estimates for PCA, multiclass LDA, CCA and SDR. Supplementary materials are available online

Statistical applications of the multivariate skew-normal distribution

Azzalini & Dalla Valle (1996) have recently discussed the multivariate skew-normal distribution which extends the class of normal distributions by the addition of a shape parameter. The first part of the present paper examines further probabilistic properties of the distribution, with special emphasis on aspects of statistical relevance. Inferential and other statistical issues are discussed in the following part, with applications to some multivariate statistics problems, illustrated by numerical examples. Finally, a further extension is described which introduces a skewing factor of an elliptical density.Comment: full-length version of the published paper, 32 pages, with 7 figures, uses psfra

Partial least squares discriminant analysis: A dimensionality reduction method to classify hyperspectral data

The recent development of more sophisticated spectroscopic methods allows acquisition of high dimensional datasets from which valuable information may be extracted using multivariate statistical analyses, such as dimensionality reduction and automatic classification (supervised and unsupervised). In this work, a supervised classification through a partial least squares discriminant analysis (PLS-DA) is performed on the hy- perspectral data. The obtained results are compared with those obtained by the most commonly used classification approaches

A D.C. Programming Approach to the Sparse Generalized Eigenvalue Problem

In this paper, we consider the sparse eigenvalue problem wherein the goal is to obtain a sparse solution to the generalized eigenvalue problem. We achieve this by constraining the cardinality of the solution to the generalized eigenvalue problem and obtain sparse principal component analysis (PCA), sparse canonical correlation analysis (CCA) and sparse Fisher discriminant analysis (FDA) as special cases. Unlike the $\ell_1$-norm approximation to the cardinality constraint, which previous methods have used in the context of sparse PCA, we propose a tighter approximation that is related to the negative log-likelihood of a Student's t-distribution. The problem is then framed as a d.c. (difference of convex functions) program and is solved as a sequence of convex programs by invoking the majorization-minimization method. The resulting algorithm is proved to exhibit \emph{global convergence} behavior, i.e., for any random initialization, the sequence (subsequence) of iterates generated by the algorithm converges to a stationary point of the d.c. program. The performance of the algorithm is empirically demonstrated on both sparse PCA (finding few relevant genes that explain as much variance as possible in a high-dimensional gene dataset) and sparse CCA (cross-language document retrieval and vocabulary selection for music retrieval) applications.Comment: 40 page

Dimension Reduction by Mutual Information Discriminant Analysis

In the past few decades, researchers have proposed many discriminant analysis (DA) algorithms for the study of high-dimensional data in a variety of problems. Most DA algorithms for feature extraction are based on transformations that simultaneously maximize the between-class scatter and minimize the withinclass scatter matrices. This paper presents a novel DA algorithm for feature extraction using mutual information (MI). However, it is not always easy to obtain an accurate estimation for high-dimensional MI. In this paper, we propose an efficient method for feature extraction that is based on one-dimensional MI estimations. We will refer to this algorithm as mutual information discriminant analysis (MIDA). The performance of this proposed method was evaluated using UCI databases. The results indicate that MIDA provides robust performance over different data sets with different characteristics and that MIDA always performs better than, or at least comparable to, the best performing algorithms.Comment: 13pages, 3 tables, International Journal of Artificial Intelligence & Application

Partial least squares discriminant analysis: A dimensionality reduction method to classify hyperspectral data

The recent development of more sophisticated spectroscopic methods allows acqui- sition of high dimensional datasets from which valuable information may be extracted using multivariate statistical analyses, such as dimensionality reduction and automatic classification (supervised and unsupervised). In this work, a supervised classification through a partial least squares discriminant analysis (PLS-DA) is performed on the hy- perspectral data. The obtained results are compared with those obtained by the most commonly used classification approaches