Linear discriminant analysis (LDA) based classifiers tend to falter in many practical settings where the training data size is smaller than, or comparable to, the number of features. As a remedy, different regularized LDA (RLDA) methods have been proposed. These methods may still perform poorly depending on the size and quality of the available training data. In particular, the test data deviation from the training data model, for example, due to noise contamination, can cause severe performance degradation. Moreover, these methods commit further to the Gaussian assumption (upon which LDA is established) to tune their regularization parameters, which may compromise accuracy when dealing with real data. To address these issues, we propose a doubly regularized LDA classifier that we denote as R2LDA. In the proposed R2LDA approach, the RLDA score function is converted into an inner product of two vectors. By substituting the expressions of the regularized estimators of these vectors, we obtain the R2LDA score function that involves two regularization parameters. To set the values of these parameters, we adopt three existing regularization techniques; the constrained perturbation regularization approach (COPRA), the bounded perturbation regularization (BPR) algorithm, and the generalized cross-validation (GCV) method. These methods are used to tune the regularization parameters based on linear estimation models, with the sample covariance matrix's square root being the linear operator. Results obtained from both synthetic and real data demonstrate the consistency and effectiveness of the proposed R2LDA approach, especially in scenarios involving test data contaminated with noise that is not observed during the training phase.
Spectrally-Corrected and Regularized Linear Discriminant Analysis for Spiked Covariance Model
School of Science, Chang Chun University（长春大学理学院）；School of Data Sciences, Zhejiang University of Finance and Economics（浙江财经大学数据科学学院）；School of Mathematics and Statistics, Northeast Normal University（东北师范大学数学与统计学院）
Wasserstein Discriminant Analysis (WDA) is a new supervised method that can improve classification of high-dimensional data by computing a suitable linear map onto a lower dimensional subspace. Following the blueprint of classical Linear Discriminant Analysis (LDA), WDA selects the projection matrix that maximizes the ratio of two quantities: the dispersion of projected points coming from different classes, divided by the dispersion of projected points coming from the same class. To quantify dispersion, WDA uses regularized Wasserstein distances, rather than cross-variance measures which have …
We propose a compressive classification framework for settings where the data dimensionality is significantly higher than the sample size. The proposed method, referred to as compressive regularized discriminant analysis (CRDA) is based on linear discriminant analysis and has the ability to select significant features by using joint-sparsity promoting hard thresholding in the discriminant rule. Since the number of features is larger than the sample size, the method also uses state-of-the-art regularized sample covariance matrix estimators. Several analysis examples on real data sets, including…
Spectrally-Corrected and Regularized QDA Classifier for Spiked Covariance Model
School of Data Sciences, Zhejiang University of Finance and Economics（浙江财经大学数据科学学院）；School of Mathematics and Statistics, Northeast Normal University（东北师范大学数学与统计学院）；Northeast Normal University（东北师范大学）
Regularized Linear Discriminant Analysis Using a Nonlinear Covariance Matrix Estimator
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；King Abdulaziz University（阿卜杜勒阿齐兹国王大学）；Center of Excellence in Intelligent Engineering Systems (CEIES)（智能工程系统卓越中心）
Quadratic discriminant analysis (QDA) is a widely used classification technique that generalizes the linear discriminant analysis (LDA) classifier to the case of distinct covariance matrices among classes. For the QDA classifier to yield high classification performance, an accurate estimation of the covariance matrices is required. Such a task becomes all the more challenging in high dimensional settings, wherein the number of observations is comparable with the feature dimension. A popular way to enhance the performance of QDA classifier under these circumstances is to regularize the covarian…
Beijing University of Posts and Telecommunications（北京邮电大学）；Missouri University of Science and Technology（密苏里科技大学）；Mininglamp Technology（明略科技）
Linear Discriminant Analysis (LDA) is a well-known technique for feature extraction and dimension reduction. The performance of classical LDA, however, significantly degrades on the High Dimension Low Sample Size (HDLSS) data for the ill-posed inverse problem. Existing approaches for HDLSS data classification typically assume the data in question are with Gaussian distribution and deal the HDLSS classification problem with regularization. However, these assumptions are too strict to hold in many emerging real-life applications, such as enabling personalized predictive analysis using Electronic…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Duke University（杜克大学）
The use of quadratic discriminant analysis (QDA) or its regularized version (R-QDA) for classification is often not recommended, due to its well-acknowledged high sensitivity to the estimation noise of the covariance matrix. This becomes all the more the case in unbalanced data settings for which it has been found that R-QDA becomes equivalent to the classifier that assigns all observations to the same class. In this paper, we propose an improved R-QDA that is based on the use of two regularization parameters and a modified bias, properly chosen to avoid inappropriate behaviors of R-QDA in unb…
The estimation of covariance matrices of multiple classes with limited training data is a difficult problem. The sample covariance matrix (SCM) is known to perform poorly when the number of variables is large compared to the available number of samples. In order to reduce the mean squared error (MSE) of the SCM, regularized (shrinkage) SCM estimators are often used. In this work, we consider regularized SCM (RSCM) estimators for multiclass problems that couple together two different target matrices for regularization: the pooled (average) SCM of the classes and the scaled identity matrix. Regu…
University of Freiburg（弗莱堡大学）；Radboud University（拉德堡德大学）；Donders Institute for Brain, Cognition and Behaviour（唐德斯大脑、认知与行为研究所）
Covariance matrices of noisy multichannel electroencephalogram time series data are hard to estimate due to high dimensionality. In brain-computer interfaces (BCI) based on event-related potentials and a linear discriminant analysis (LDA) for classification, the state of the art to address this problem is by shrinkage regularization. We propose a novel idea to tackle this problem by enforcing a block-Toeplitz structure for the covariance matrix of the LDA, which implements an assumption of signal stationarity in short time windows for each channel. On data of 213 subjects collected under 13 ev…
East China Normal University（华东师范大学）；Shanghai Jiao Tong University（上海交通大学）
Acoustic anomaly detection aims at distinguishing abnormal acoustic signals from the normal ones. It suffers from the class imbalance issue and the lacking in the abnormal instances. In addition, collecting all kinds of abnormal or unknown samples for training purpose is impractical and timeconsuming. In this paper, a novel Gaussian Mixture Generative Adversarial Network (GMGAN) is proposed under semi-supervised learning framework, in which the underlying structure of training data is not only captured in spectrogram reconstruction space, but also can be further restricted in the space of late…
Technicolor AI Lab（特艺集团人工智能实验室）；Marshall School of Business, University of Southern California（南加州大学马歇尔商学院）；University of Washington（华盛顿大学）
Prior knowledge on properties of a target model often come as discrete or combinatorial descriptions. This work provides a unified computational framework for defining norms that promote such structures. More specifically, we develop associated tools for optimization involving such norms given only the orthogonal projection oracle onto the non-convex set of desired models. As an example, we study a norm, which we term the doubly-sparse norm, for promoting vectors with few nonzero entries taking only a few distinct values. We further discuss how the K-means algorithm can serve as the underlying…
In few-shot classification, the primary goal is to learn representations from a few samples that generalize well for novel classes. In this paper, we propose an efficient low displacement rank (LDR) regularization strategy termed Ortho-Shot; a technique that imposes orthogonal regularization on the convolutional layers of a few-shot classifier, which is based on the doubly-block toeplitz (DBT) matrix structure. The regularized convolutional layers of the few-shot classifier enhances model generalization and intra-class feature embeddings that are crucial for few-shot learning. Overfitting is a…
