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

Benchmarking least squares support vector machine classifiers.

In Support Vector Machines (SVMs), the solution of the classification problem is characterized by a ( convex) quadratic programming (QP) problem. In a modified version of SVMs, called Least Squares SVM classifiers (LS-SVMs), a least squares cost function is proposed so as to obtain a linear set of equations in the dual space. While the SVM classifier has a large margin interpretation, the LS-SVM formulation is related in this paper to a ridge regression approach for classification with binary targets and to Fisher's linear discriminant analysis in the feature space. Multiclass categorization problems are represented by a set of binary classifiers using different output coding schemes. While regularization is used to control the effective number of parameters of the LS-SVM classifier, the sparseness property of SVMs is lost due to the choice of the 2-norm. Sparseness can be imposed in a second stage by gradually pruning the support value spectrum and optimizing the hyperparameters during the sparse approximation procedure. In this paper, twenty public domain benchmark datasets are used to evaluate the test set performance of LS-SVM classifiers with linear, polynomial and radial basis function (RBF) kernels. Both the SVM and LS-SVM classifier with RBF kernel in combination with standard cross-validation procedures for hyperparameter selection achieve comparable test set performances. These SVM and LS-SVM performances are consistently very good when compared to a variety of methods described in the literature including decision tree based algorithms, statistical algorithms and instance based learning methods. We show on ten UCI datasets that the LS-SVM sparse approximation procedure can be successfully applied.least squares support vector machines; multiclass support vector machines; sparse approximation; discriminant-analysis; sparse approximation; learning algorithms; classification; framework; kernels; time; SISTA;

Scalable Randomized Kernel Methods for Multiview Data Integration and Prediction

We develop scalable randomized kernel methods for jointly associating data from multiple sources and simultaneously predicting an outcome or classifying a unit into one of two or more classes. The proposed methods model nonlinear relationships in multiview data together with predicting a clinical outcome and are capable of identifying variables or groups of variables that best contribute to the relationships among the views. We use the idea that random Fourier bases can approximate shift-invariant kernel functions to construct nonlinear mappings of each view and we use these mappings and the outcome variable to learn view-independent low-dimensional representations. Through simulation studies, we show that the proposed methods outperform several other linear and nonlinear methods for multiview data integration. When the proposed methods were applied to gene expression, metabolomics, proteomics, and lipidomics data pertaining to COVID-19, we identified several molecular signatures forCOVID-19 status and severity. Results from our real data application and simulations with small sample sizes suggest that the proposed methods may be useful for small sample size problems. Availability: Our algorithms are implemented in Pytorch and interfaced in R and would be made available at: https://github.com/lasandrall/RandMVLearn.Comment: 24 pages, 5 figures, 4 table