On the optimal class representation in linear discriminant analysis

The Geometry of Nonlinear Embeddings in Kernel Discriminant Analysis

Fisher's linear discriminant analysis is a classical method for classification, yet it is limited to capturing linear features only. Kernel discriminant analysis as an extension is known to successfully alleviate the limitation through a nonlinear feature mapping. We study the geometry of nonlinear embeddings in discriminant analysis with polynomial kernels and Gaussian kernel by identifying the population-level discriminant function that depends on the data distribution and the kernel. In order to obtain the discriminant function, we solve a generalized eigenvalue problem with between-class and within-class covariance operators. The polynomial discriminants are shown to capture the class difference through the population moments explicitly. For approximation of the Gaussian discriminant, we use a particular representation of the Gaussian kernel by utilizing the exponential generating function for Hermite polynomials. We also show that the Gaussian discriminant can be approximated using randomized projections of the data. Our results illuminate how the data distribution and the kernel interact in determination of the nonlinear embedding for discrimination, and provide a guideline for choice of the kernel and its parameters.

Supervised Domain Adaptation: A Graph Embedding Perspective and a Rectified Experimental Protocol

Domain Adaptation is the process of alleviating distribution gaps between data from different domains. In this paper, we show that Domain Adaptation methods using pair-wise relationships between source and target domain data can be formulated as a Graph Embedding in which the domain labels are incorporated into the structure of the intrinsic and penalty graphs. Specifically, we analyse the loss functions of three existing state-of-the-art Supervised Domain Adaptation methods and demonstrate that they perform Graph Embedding. Moreover, we highlight some generalisation and reproducibility issues related to the experimental setup commonly used to demonstrate the few-shot learning capabilities of these methods. To assess and compare Supervised Domain Adaptation methods accurately, we propose a rectified evaluation protocol, and report updated benchmarks on the standard datasets Office31 (Amazon, DSLR, and Webcam), Digits (MNIST, USPS, SVHN, and MNIST-M) and VisDA (Synthetic, Real).

Self-weighted Robust LDA for Multiclass Classification with Edge Classes

Linear discriminant analysis (LDA) is a popular technique to learn the most discriminative features for multi-class classification. A vast majority of existing LDA algorithms are prone to be dominated by the class with very large deviation from the others, i.e., edge class, which occurs frequently in multi-class classification. First, the existence of edge classes often makes the total mean biased in the calculation of between-class scatter matrix. Second, the exploitation of ℓ 2 -norm based between-class distance criterion magnifies the extremely large distance corresponding to edge class. In this regard, a novel self-weighted robust LDA with ℓ 2,1 -norm based pairwise between-class distance criterion, called SWRLDA, is proposed for multi-class classification especially with edge classes. SWRLDA can automatically avoid the optimal mean calculation and simultaneously learn adaptive weights for each class pair without setting any additional parameter. An efficient re-weighted algorithm is exploited to derive the global optimum of the challenging ℓ 2,1 -norm maximization problem. The proposed SWRLDA is easy to implement and converges fast in practice. Extensive experiments demonstrate that SWRLDA performs favorably against other compared methods on both synthetic and real-world datasets while presenting superior computational efficiency in comparison with other techniques.

A Machine Learning-Based Approach for the Prediction of Acute Coronary Syndrome Requiring Revascularization

The aim of this study is to predict acute coronary syndrome (ACS) requiring revascularization in those patients presenting early-stage angina-like symptom using machine learning algorithms. We obtained data from 2344 ACS patients, who required revascularization and from 3538 non-ACS patients. We analyzed 20 features that are relevant to ACS using standard algorithms, support vector machines and linear discriminant analysis. Based on feature pattern and filter characteristics, we analyzed and extracted a strong prediction function out of the 20 selected features. The obtained prediction functions are relevant showing the area under curve of 0.860 for the prediction of ACS that requiring revascularization. Some features are missing in many data though they are considered to be very informative; it turned out that omitting those features from the input and using more data without those features for training improves the prediction accuracy. Additionally, from the investigation using the receiver operating characteristic curves, a reliable prediction of 2.60% of non-ACS patients could be made with a specificity of 1.0. For those 2.60% non-ACS patients, we can consider the recommendation of medical treatment without risking misdiagnosis of the patients requiring revascularization. We investigated prediction algorithm to select ACS patients requiring revascularization and non-ACS patients presenting angina-like symptoms at an early stage. In the future, a large cohort study is necessary to increase the prediction accuracy and confirm the possibility of safely discriminating the non-ACS patients from the ACS patients with confidence.

DBSDA : Lowering the Bound of Misclassification Rate for Sparse Linear Discriminant Analysis via Model Debiasing

Linear discriminant analysis (LDA) is a well-known technique for linear classification, feature extraction, and dimension reduction. To improve the accuracy of LDA under the high dimension low sample size (HDLSS) settings, shrunken estimators, such as Graphical Lasso, can be used to strike a balance between biases and variances. Although the estimator with induced sparsity obtains a faster convergence rate, however, the introduced bias may also degrade the performance. In this paper, we theoretically analyze how the sparsity and the convergence rate of the precision matrix (also known as inverse covariance matrix) estimator would affect the classification accuracy by proposing an analytic model on the upper bound of an LDA misclassification rate. Guided by the model, we propose a novel classifier, DBSDA , which improves classification accuracy through debiasing. Theoretical analysis shows that DBSDA possesses a reduced upper bound of misclassification rate and better asymptotic properties than sparse LDA (SDA). We conduct experiments on both synthetic datasets and real application datasets to confirm the correctness of our theoretical analysis and demonstrate the superiority of DBSDA over LDA, SDA, and other downstream competitors under HDLSS settings.

Generalized Multi-View Embedding for Visual Recognition and Cross-Modal Retrieval

In this paper, the problem of multi-view embedding from different visual cues and modalities is considered. We propose a unified solution for subspace learning methods using the Rayleigh quotient, which is extensible for multiple views, supervised learning, and nonlinear embeddings. Numerous methods including canonical correlation analysis, partial least square regression, and linear discriminant analysis are studied using specific intrinsic and penalty graphs within the same framework. Nonlinear extensions based on kernels and (deep) neural networks are derived, achieving better performance than the linear ones. Moreover, a novel multi-view modular discriminant analysis is proposed by taking the view difference into consideration. We demonstrate the effectiveness of the proposed multi-view embedding methods on visual object recognition and cross-modal image retrieval, and obtain superior results in both applications compared to related methods.

Heteroscedastic Max-min Distance Analysis for Dimensionality Reduction

Max-min distance analysis (MMDA) performs dimensionality reduction by maximizing the minimum pairwise distance between classes in the latent subspace under the homoscedastic assumption, which can address the class separation problem caused by the Fisher criterion, but is incapable of tackling heteroscedastic data properly. In this paper, we propose two heteroscedastic MMDA (HMMDA) methods to employ the differences of class covariances. Whitened HMMDA (WHMMDA) extends MMDA by utilizing the Chernoff distance as the separability measure between classes in the whitened space. Orthogonal HMMDA (OHMMDA) incorporates the maximization of the minimal pairwise Chernoff distance and the minimization of class compactness into a trace quotient formulation with an orthogonal constraint of the transformation, which can be solved by bisection search. Two variants of OHMMDA further encode the margin information by using only neighboring samples to construct the intra-class and inter-class scatters. Experiments on several UCI datasets and two face databases demonstrate the effectiveness of the HMMDA methods.

Fluid Dynamic Models for Bhattacharyya-Based Discriminant Analysis

Classical discriminant analysis attempts to discover a low-dimensional subspace where class label information is maximally preserved under projection. Canonical methods for estimating the subspace optimize an information-theoretic criterion that measures the separation between the class-conditional distributions. Unfortunately, direct optimization of the information-theoretic criteria is generally non-convex and intractable in high-dimensional spaces. In this work, we propose a novel, tractable algorithm for discriminant analysis that considers the class-conditional densities as interacting fluids in the high-dimensional embedding space. We use the Bhattacharyya criterion as a potential function that generates forces between the interacting fluids, and derive a computationally tractable method for finding the low-dimensional subspace that optimally constrains the resulting fluid flow. We show that this model properly reduces to the optimal solution for homoscedastic data as well as for heteroscedastic Gaussian distributions with equal means. We also extend this model to discover optimal filters for discriminating Gaussian processes and provide experimental results and comparisons on a number of datasets.

Class-Specific Kernel Discriminant Analysis Revisited: Further Analysis and Extensions

In this paper, we revisit class-specific kernel discriminant analysis (KDA) formulation, which has been applied in various problems, such as human face verification and human action recognition. We show that the original optimization problem solved for the determination of class-specific discriminant projections is equivalent to a low-rank kernel regression (LRKR) problem using training data-independent target vectors. In addition, we show that the regularized version of class-specific KDA is equivalent to a regularized LRKR problem, exploiting the same targets. This analysis allows us to devise a novel fast solution. Furthermore, we devise novel incremental, approximate and deep (hierarchical) variants. The proposed methods are tested in human facial image and action video verification problems, where their effectiveness and efficiency is shown.

Symmetric subspace learning for image analysis

Subspace learning (SL) is one of the most useful tools for image analysis and recognition. A large number of such techniques have been proposed utilizing a priori knowledge about the data. In this paper, new subspace learning techniques are presented that use symmetry constraints in their objective functions. The rational behind this idea is to exploit the a priori knowledge that geometrical symmetry appears in several types of data, such as images, objects, faces, and so on. Experiments on artificial, facial expression recognition, face recognition, and object categorization databases highlight the superiority and the robustness of the proposed techniques, in comparison with standard SL techniques.