Regularized discriminant analysis for the small sample size problem in face recognition

Linear Discriminant Analysis (LDA) has been one of the popular subspace methods for face recognition. But this method suffers from the small sample size (SSS) problem, also known as 'curse of dimensionality'. Various techniques have been proposed in literature to overcome this limitation. But it is still unclear which method provides the best solution to SSS problem. In this paper, we have investigated the performance of some popular subspace methods such as principal component analysis (PCA), PCA + LDA, LDA via QR decomposition, Null-space LDA, Exponential Discriminant Analysis (EDA), PCA+EDA etc. Extensive experiments have been performed on five publically available face datasets viz. AR, CMU-PIE, PIX, Yale and YaleB. The performance is measured in terms of average classification accuracy. Experimental results show that the performance increases with the increase in the number of images per person in training set irrespective of the datasets. There is no clear winner among the subspace methods under investigation. But, the performance of PCA+LDA and PCA+EDA is consistent in tackling SSS problem irrespective of the dataset and can also handle the illumination variation in face recognition.

This paper presents a novel and effective method for facial expression recognition including happiness, disgust, fear, anger, sadness, surprise, and neutral state. The proposed method utilizes a regularized discriminant analysis-based boosting algorithm (RDAB) with effective Gabor features to recognize the facial expressions. Entropy criterion is applied to select the effective Gabor feature which is a subset of informative and nonredundant Gabor features. The proposed RDAB algorithm uses RDA as a learner in the boosting algorithm. The RDA combines strengths of linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA). It solves the small sample size and ill-posed problems suffered from QDA and LDA through a regularization technique. Additionally, this study uses the particle swarm optimization (PSO) algorithm to estimate optimal parameters in RDA. Experiment results demonstrate that our approach can accurately and robustly recognize facial expressions.

Face recognition using direct, weighted linear discriminant analysis and modular subspaces

Linear Discriminant Analysis (LDA) is a popular statistical method for both feature extraction and dimensionality reduction in face recognition. The major drawback of LDA is the so-called small sample size (3S) problem. This problem always occurs when the total number of training samples is smaller than the dimension of feature space. Under this situation, the within-class scatter matrix Sw becomes singular and LDA approach cannot be implemented directly. To overcome the 3S problem, this paper proposes a novel wavelet-face based subspace LDA algorithm. Wavelet-face feature extraction and dimensionality reduction are based on two-level D4-filter wavelet transform and discarding the null space of total class scatter matrix St. It is shown that our obtained projection matrix satisfies the uncorrelated constraint conditions. Hence in the sense of statistical uncorrelation, this projection matrix is optimal. The proposed method for face recognition has been evaluated with two public available databases, namely ORL and FERET databases. Comparing with existing LDA-based methods to solve the 3S problem, our method gives the best performance.

Face recognition using direct, weighted linear discriminant analysis and modular subspaces

Kernel quadratic discriminant analysis for small sample size problem

Sufficient dimension-reduction methods provide effective ways to visualize discriminant analysis problems. For example, Cook and Yin showed that the dimension-reduction method of sliced average variance estimation (SAVE) identifies variates that are equivalent to a quadratic discriminant analysis (QDA) solution. This article makes this connection explicit to motivate the use of SAVE variates in exploratory graphics for discriminant analysis. Classification can then be based on the SAVE variates using a suitable distance measure. If the chosen measure is Mahalanobis distance, then classification is identical to QDA using the original variables. Just as canonical variates provide a useful way to visualize linear discriminant analysis (LDA), so do SAVE variates help visualize QDA. This would appear to be particularly useful given the lack of graphical tools for QDA in current software. Furthermore, whereas LDA and QDA can be sensitive to nonnormality, SAVE is more robust.

Many contemporary studies involve the classification of a subject into two classes based on n observations of the p variables associated with the subject. Under the assumption that the variables are normally distributed, the well-known linear discriminant analysis (LDA) assumes a common covariance matrix over the two classes while the quadratic discriminant analysis (QDA) allows different covariance matrices. When p is much smaller than n, even if they both diverge, the LDA and QDA have the smallest asymptotic misclassification rates for the cases of equal and unequal covariance matrices, respectively. However, modern statistical studies often face classification problems with the number of variables much larger than the sample size n, and the classical LDA and QDA can perform poorly. In fact, we give an example in which the QDA performs as poorly as random guessing even if we know the true covariances. Under some sparsity conditions on the unknown means and covariance matrices of the two classes, we propose a sparse QDA based on thresholding that has the smallest asymptotic misclassification rate conditional on the training data. We discuss an example of classifying normal and tumor colon tissues based on a set of p = 2; 000 genes and a sample of size n = 62, and another example of a cardiovascular study for n = 222 subjects with p = 2; 434 genes. A simulation is also conducted to check the performance of the proposed method.

In this paper, an adaptive approximation image reconstruction method based on orthogonal triangular with column pivoting (QRCP) decomposition algorithm is proposed for single sample problem in face recognition. By using QRCP the single sample and its transpose are decomposed to two sets of basis images. Then an adaptive approximation image reconstruction method is proposed to reconstruct two approximation images from the two basis image sets respectively. The single training sample and its two approximation images of each object form a new training set, which can make the fisher linear discriminant analysis (FLDA) be applied to single sample problem in face recognition. The performance of the proposed method is verified on Yale, FERET, and ORL face databases. The experimental results indicate that the proposed method is efficient and outperforms some existing methods which are proposed to overcome the single sample problem.

The objective of this study is to concentrate on the classification method of the logistic regression and the discriminant analysis by using the simulation dataset and the liver patients as the actual data. These datasets are used the binary dependent variable depending on the correlated independent variables or called multicollinearity data. The standard classification method is logistic regression, which uses the logit function’s probability to conduct the dichotomous dependent variable. The iteration process can be solved to estimate logit function parameters and explain the relationship between a dependent binary variable and independent variables. Discriminant analysis is a powerful classification based on linear discriminant analysis (LDA), quadratic discriminant analysis (QDA), and regularized discriminant analysis (RDA). These methods consider the decision boundaries by building a classifier model on the multivariate normal distribution. LDA defines the standard covariance matrix, but QDA has an individual covariance matrix. RDA extends from QDA by setting the regularized parameter to estimate the covariance matrix. In the case of the simulation study, the independent variables are generated by defining the constant correlation on the multivariate normal distribution that made the multicollinearity problem. Then the binary response variable can be approximated from the logit function. For application to actual data, we expressed the classification of type liver and non-liver patients as the dependent variables and obtained patient personal information on the nine independent variables. The highest average percentage of accuracy determines the performance of these methods. The results have shown that the logistic regression was successful when using small independent variables, but the RDA performed when using large independent variables.

This paper addresses two LDA problems in face recognition. The first one is small sample size (S3) problem while the second is illumination and pose variations. To overcome the S3 problem, this paper proposes a new method in subspace approach in determining the optimal projection for LDA. Also, an in-depth investigation is conducted on the influence of different illuminations and poses variations. Comparisons with existing LDA-based methods are performed using FERET and Yale Group B face databases. The experimental results show that the proposed method gives the best performance comparing with the existing LDA-based methods for both databases. Moreover, the computational cost of the proposed method is near the same as the existing fastest LDA-based method.

The performance of seven classification methods, binary logistic (BLR), probit (PR) and cumulative probit (CPR) regression, linear (LDA) and quadratic (QDA) discriminant analysis, artificial neural networks (ANN), and naïve Bayes classification (NBC), is examined in skeletal sex estimation. These methods were tested using cranial and pelvic sexually dimorphic traits recorded on a modern documented collection, the Athens Collection. For their implementation, an R package has been written to perform cross-validated (CV) sex classification and give the discriminant function of each of the methods studied. A simple algorithm that combines two discriminant functions is also proposed. It was found that the differences in the classification performance between BLR, PR, CPR, LDA, QDA, ANN, and NBC are overall small. However, LDA is simpler and more flexible than CPR, QDA, and ANN and has a small but clear advantage over BLR, NBC, and PR. Consequently, LDA may be preferred in skeletal sex estimation. Finally, it is striking that the combination of pelvic and cranial traits via their discriminant functions, determined either by BLR or LDA, removes practically any population-specificity and yields much better predictions than the individual functions; in fact, the prediction accuracy increases above 97%.

In the biometrics community, face and speaker recognition are mature fields in which several systems have been proposed over the past twenty years. While existing systems perform well under controlled recording conditions, mismatch caused by the use of different sensors or a lack of cooperation from the subject still significantly affects performance, especially in challenging scenarios such as in forensics. Furthermore, existing methods suffer from scalability issues, which prevents them from taking advantage of increasingly large amounts of training data. This is otherwise a promising approach to improve accuracy in such challenging scenarios. In this thesis we address these problems of mismatch and complexity by developing scalable probabilistic models that we apply to face, speaker and bimodal recognition. Our contributions are four-fold. First, we propose a unified framework for session variability modeling techniques based on Gaussian mixture models (GMM), that encompasses inter-session variability (ISV) modeling, joint factor analysis (JFA) and total variability (TV) modeling. Second, we propose a novel exact and scalable formulation of probabilistic linear discriminant analysis (PLDA), which is a probabilistic and generative framework that models between-class and within-class variations. This formulation solves a major scalability issue, by improving both the time complexity of the training procedure from cubic to linear with respect to the number of samples per class, and the complexity of the scoring procedure. Furthermore, the implementations of all the proposed techniques are integrated into a novel collaborative open source software library called Bob 1 that enforces fair evaluations and encourages reproducible research. Fourth and finally, large-scale experiments are conducted with all of the above machine learning algorithms on several databases such as FRGC for face recognition, NIST SRE12 for speaker recognition and MOBIO for bimodal recognition, showing competitive performances.

Linear and Quadratic Discriminant Analysis have been used widely in many areas of data mining, machine learning, and bioinformatics. Friedman proposed a compromise between Linear and Quadratic Discriminant Analysis, called Regularized Discriminant Analysis (RDA), which has been shown to be more flexible in dealing with various class distributions. RDA applies the regularization techniques by employing two regularization parameters, which are chosen to jointly maximize the classification performance. The optimal pair of parameters is commonly estimated via cross-validation from a set of candidate pairs. It is computationally prohibitive for high dimensional data, especially when the candidate set is large, which limits the applications of RDA to low dimensional data.In this paper, a novel algorithm for RDA is presented for high dimensional data. It can estimate the optimal regularization parameters from a large set of parameter candidates efficiently. Experiments on a variety of datasets confirm the claimed theoretical estimate of the efficiency, and also show that, for a properly chosen pair of regularization parameters, RDA performs favorably in classification, in comparison with other existing classification methods.

Lung cancer is a type of cancer that starts from the lungs with abnormal cell growth. Lung cancer is a type of cancer with the largest number of sufferers. In this research, Regularized Discriminant Analysis (RDA) analysis is used to classify the types of lung cancer that exist. Regularized Discriminant Analysis (RDA) is a generalization of Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA). If the alpha parameter is set to 1, this operator performs an LDA. Likewise if the alpha parameter is set to 0, this operator performs QDA. Discriminant analysis is used to determine which variables distinguish between two or more groups that occur naturally. With an accuracy value of 60%, it can be ascertained the type of lung cancer suffered by patients for ease of care and treatment.