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 unbalanced settings and to ensure the best possible classification performance. The design of the proposed classifier builds on a refined asymptotic analysis of its performance when the number of samples and that of features grow large simultaneously, which allows to cope efficiently with the high-dimensionality frequently met within the big data paradigm. The performance of the proposed classifier is assessed on both real and synthetic data sets and was shown to be much better than what one would expect from a traditional R-QDA.
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…
TU Eindhoven（埃因霍温理工大学）；TU Delft（代尔夫特理工大学）；University of Copenhagen（哥本哈根大学）
In practice, the data distribution at test time often differs, to a smaller or larger extent, from that of the original training data. Consequentially, the so-called source classifier, trained on the available labelled data, deteriorates on the test, or target, data. Domain adaptive classifiers aim to combat this problem, but typically assume some particular form of domain shift. Most are not robust to violations of domain shift assumptions and may even perform worse than their non-adaptive counterparts. We construct robust parameter estimators for discriminant analysis that guarantee performa…
Technische Universität Kaiserslautern（凯泽斯劳滕工业大学）；Helwan University（赫尔万大学）
Nested Cavity Classifier (NCC) is a classification rule that pursues partitioning the feature space, in parallel coordinates, into convex hulls to build decision regions. It is claimed in some literatures that this geometric-based classifier is superior to many others, particularly in higher dimensions. First, we give an example on how NCC can be inefficient, then motivate a remedy by combining the NCC with the Linear Discriminant Analysis (LDA) classifier. We coin the term Nested Cavity Discriminant Analysis (NCDA) for the resulting classifier. Second, a simulation study is conducted to compa…
This paper deals with the binary classification task when the target class has the lower probability of occurrence. In such situation, it is not possible to build a powerful classifier by using standard methods such as logistic regression, classification tree, discriminant analysis, etc. To overcome this short-coming of these methods which yield classifiers with low sensibility, we tackled the classification problem here through an approach based on the association rules learning. This approach has the advantage of allowing the identification of the patterns that are well correlated with the t…
The University of the Basque Country UPV/EHU（巴斯克大学）；Basque Center for Applied Mathematics BCAM（巴斯克应用数学中心）
Although a great methodological effort has been invested in proposing competitive solutions to the class-imbalance problem, little effort has been made in pursuing a theoretical understanding of this matter. In order to shed some light on this topic, we perform, through a novel framework, an exhaustive analysis of the adequateness of the most commonly used performance scores to assess this complex scenario. We conclude that using unweighted Hölder means with exponent $p \leq 1$ to average the recalls of all the classes produces adequate scores which are capable of determining whether a classif…
We consider the two-group classification problem and propose a kernel classifier based on the optimal scoring framework. Unlike previous approaches, we provide theoretical guarantees on the expected risk consistency of the method. We also allow for feature selection by imposing structured sparsity using weighted kernels. We propose fully-automated methods for selection of all tuning parameters, and in particular adapt kernel shrinkage ideas for ridge parameter selection. Numerical studies demonstrate the superior classification performance of the proposed approach compared to existing nonparam…
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）
COMSATS University Islamabad（COMSATS大学伊斯兰堡校区）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）
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 e…
This article carries out a large dimensional analysis of standard regularized discriminant analysis classifiers designed on the assumption that data arise from a Gaussian mixture model with different means and covariances. The analysis relies on fundamental results from random matrix theory (RMT) when both the number of features and the cardinality of the training data within each class grow large at the same pace. Under mild assumptions, we show that the asymptotic classification error approaches a deterministic quantity that depends only on the means and covariances associated with each clas…
Department of Engineering, ECE, Aarhus University（丹麦奥胡斯大学工程系（ECE））
In this paper we formulate a probabilistic model for class-specific discriminant subspace learning. The proposed model can naturally incorporate the multi-modal structure of the negative class, which is neglected by existing class-specific methods. Moreover, it can be directly used to define a class-specific probabilistic classification rule in the discriminant subspace. We show that existing class-specific discriminant analysis methods are special cases of the proposed probabilistic model and, by casting them as probabilistic models, they can be extended to class-specific classifiers. We illu…
In the context of Multi Instance Learning, we analyze the Single Instance (SI) learning objective. We show that when the data is unbalanced and the family of classifiers is sufficiently rich, the SI method is a useful learning algorithm. In particular, we show that larger data imbalance, a quality that is typically perceived as negative, in fact implies a better resilience of the algorithm to the statistical dependencies of the objects in bags. In addition, our results shed new light on some known issues with the SI method in the setting of linear classifiers, and we show that these issues are…
Regularized discriminant analysis (RDA), proposed by Friedman (1989), is a widely popular classifier that lacks interpretability and is impractical for high-dimensional data sets. Here, we present an interpretable and computationally efficient classifier called high-dimensional RDA (HDRDA), designed for the small-sample, high-dimensional setting. For HDRDA, we show that each training observation, regardless of class, contributes to the class covariance matrix, resulting in an interpretable estimator that borrows from the pooled sample covariance matrix. Moreover, we show that HDRDA is equivale…
Indian Statistical Institute（印度统计研究所）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Quadratic discriminant analysis (QDA) is a widely used statistical tool to classify observations from different multivariate Normal populations. The generalized quadratic discriminant analysis (GQDA) classification rule/classifier, which generalizes the QDA and the minimum Mahalanobis distance (MMD) classifiers to discriminate between populations with underlying elliptically symmetric distributions competes quite favorably with the QDA classifier when it is optimal and performs much better when QDA fails under non-Normal underlying distributions, e.g. Cauchy distribution. However, the classifi…
Department of Mathematics, Nazarbayev University（纳扎尔拜耶夫大学数学系）；Machine Learning Department, Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学机器学习系）；Department of Mathematical Sciences, Purdue University Fort Wayne（普渡大学福恩湾分校数学科学系）；Remote Sensing Department, National Center of Space Research and Technology（国家空间研究与技术中心遥感部）
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Effat University（Effat大学）
Datasets from the fields of bioinformatics, chemometrics, and face recognition are typically characterized by small samples of high-dimensional data. Among the many variants of linear discriminant analysis that have been proposed in order to rectify the issues associated with classification in such a setting, the classifier in [1], composed of an ensemble of randomly projected linear discriminants, seems especially promising; it is computationally efficient and, with the optimal projection dimension parameter setting, is competitive with the state-of-the-art. In this work, we seek to further u…
Large-sample data became prevalent as data acquisition became cheaper and easier. While a large sample size has theoretical advantages for many statistical methods, it presents computational challenges. Sketching, or compression, is a well-studied approach to address these issues in regression settings, but considerably less is known about its performance in classification settings. Here we consider the computational issues due to large sample size within the discriminant analysis framework. We propose a new compression approach for reducing the number of training samples for linear and quadra…
Delft University of Technology（代尔夫特理工大学）；University of Copenhagen（哥本哈根大学）；Leiden University Medical Center（莱顿大学医学中心）
Semi-supervised learning is an important and active topic of research in pattern recognition. For classification using linear discriminant analysis specifically, several semi-supervised variants have been proposed. Using any one of these methods is not guaranteed to outperform the supervised classifier which does not take the additional unlabeled data into account. In this work we compare traditional Expectation Maximization type approaches for semi-supervised linear discriminant analysis with approaches based on intrinsic constraints and propose a new principled approach for semi-supervised l…
Interval Fisher's Discriminant Analysis and Visualisation
Dep. of Mathematics, Instituto Superior Técnico, ULisboa, Portugal（数学系，技术高级学院，里斯本大学，葡萄牙）；CEMAT, Instituto Superior Técnico, ULisboa, Portugal（CEMAT，技术高级学院，里斯本大学，葡萄牙）；CAMGSD and Dep. of Mathematics, Instituto Superior Técnico, ULisboa, Portugal（CAMGSD和数学系，技术高级学院，里斯本大学，葡萄牙）
Machine learning techniques help to understand patterns of a dataset to create a defense mechanism against cyber attacks. However, it is difficult to construct a theoretical model due to the imbalances in the dataset for discriminating attacks from the overall dataset. Multilayer Perceptron (MLP) technique will provide improvement in accuracy and increase the performance of detecting the attack and benign data from a balanced dataset. We have worked on the UGR'16 dataset publicly available for this work. Data wrangling has been done due to prepare test set from in the original set. We fed the …
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）
We develop a class of rules spanning the range between quadratic discriminant analysis and naive Bayes, through a path of sparse graphical models. A group lasso penalty is used to introduce shrinkage and encourage a similar pattern of sparsity across precision matrices. It gives sparse estimates of interactions and produces interpretable models. Inspired by the connected-components structure of the estimated precision matrices, we propose the community Bayes model, which partitions features into several conditional independent communities and splits the classification problem into separate sma…
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…
Google Research（谷歌研究院）；Google Brain（谷歌大脑）；Stanford University（斯坦福大学）
Knowing when a classifier's prediction can be trusted is useful in many applications and critical for safely using AI. While the bulk of the effort in machine learning research has been towards improving classifier performance, understanding when a classifier's predictions should and should not be trusted has received far less attention. The standard approach is to use the classifier's discriminant or confidence score; however, we show there exists an alternative that is more effective in many situations. We propose a new score, called the trust score, which measures the agreement between the …
Electrical and Computer Engineering Program, King Abdullah University of Science and Technology（计算机与电气工程系，沙特国王阿卜杜拉科技大学）；Center of Excellence for NEOM Research at KAUST, King Abdullah University of Science and Technology（沙特国王阿卜杜拉科技大学NEOM研究卓越中心）
Unlike its intercept, a linear classifier's weight vector cannot be tuned by a simple grid search. Hence, this paper proposes weight vector tuning of a generic binary linear classifier through the parameterization of a decomposition of the discriminant by a scalar which controls the trade-off between conflicting informative and noisy terms. By varying this parameter, the original weight vector is modified in a meaningful way. Applying this method to a number of linear classifiers under a variety of data dimensionality and sample size settings reveals that the classification performance loss du…
Manifold learning is used for dimensionality reduction, with the goal of finding a projection subspace to increase and decrease the inter- and intraclass variances, respectively. However, a bottleneck for subspace learning methods often arises from the high dimensionality of datasets. In this paper, a hierarchical approach is proposed to scale subspace learning methods, with the goal of improving classification in large datasets by a range of 3% to 10%. Different combinations of methods are studied. We assess the proposed method on five publicly available large datasets, for different eigen-va…
Optimisation and Logistics Group, The University of Adelaide（阿德莱德大学优化与物流团队）
A classification algorithm, called the Linear Centralization Classifier (LCC), is introduced. The algorithm seeks to find a transformation that best maps instances from the feature space to a space where they concentrate towards the center of their own classes, while maximimizing the distance between class centers. We formulate the classifier as a quadratic program with quadratic constraints. We then simplify this formulation to a linear program that can be solved effectively using a linear programming solver (e.g., simplex-dual). We extend the formulation for LCC to enable the use of kernel f…
Auburn University（奥本大学）；New Jersey Institute of Technology（新泽西理工学院）
We propose a new approach, called as functional deep neural network (FDNN), for classifying multi-dimensional functional data. Specifically, a deep neural network is trained based on the principle components of the training data which shall be used to predict the class label of a future data function. Unlike the popular functional discriminant analysis approaches which rely on Gaussian assumption, the proposed FDNN approach applies to general non-Gaussian multi-dimensional functional data. Moreover, when the log density ratio possesses a locally connected functional modular structure, we show …
Mid Sweden University（中瑞典大学）；Swedish University of Agricultural Sciences（瑞典农业科学大学）
We present a growing dimension asymptotic formalism. The perspective in this paper is classification theory and we show that it can accommodate probabilistic networks classifiers, including naive Bayes model and its augmented version. When represented as a Bayesian network these classifiers have an important advantage: The corresponding discriminant function turns out to be a specialized case of a generalized additive model, which makes it possible to get closed form expressions for the asymptotic misclassification probabilities used here as a measure of classification accuracy. Moreover, in t…
We propose a modification of linear discriminant analysis, referred to as compressive regularized discriminant analysis (CRDA), for analysis of high-dimensional datasets. CRDA is specially designed for feature elimination purpose and can be used as gene selection method in microarray studies. CRDA lends ideas from $\ell_{q,1}$ norm minimization algorithms in the multiple measurement vectors (MMV) model and utilizes joint-sparsity promoting hard thresholding for feature elimination. A regularization of the sample covariance matrix is also needed as we consider the challenging scenario where the…
In supervised classification problems, the test set may contain data points belonging to classes not observed in the learning phase. Moreover, the same units in the test data may be measured on a set of additional variables recorded at a subsequent stage with respect to when the learning sample was collected. In this situation, the classifier built in the learning phase needs to adapt to handle potential unknown classes and the extra dimensions. We introduce a model-based discriminant approach, Dimension-Adaptive Mixture Discriminant Analysis (D-AMDA), which can detect unobserved classes and a…
Classification is an important tool with many useful applications. Among the many classification methods, Fisher's Linear Discriminant Analysis (LDA) is a traditional model-based approach which makes use of the covariance information. However, in the high-dimensional, low-sample size setting, LDA cannot be directly deployed because the sample covariance is not invertible. While there are modern methods designed to deal with high-dimensional data, they may not fully use the covariance information as LDA does. Hence in some situations, it is still desirable to use a model-based method such as LD…
University of Insubria（因苏布里亚大学）；University of Essex（埃塞克斯大学）
In this paper, we propose a design methodology for one-class classifiers using an ensemble-of-classifiers approach. The objective is to select the best structures created during the training phase using an ensemble of spanning trees. It takes the best classifier, partitioning the area near a pattern into $γ^{γ-2}$ sub-spaces and combining all possible spanning trees that can be created starting from $γ$ nodes. The proposed method leverages on a supervised classification methodology and the concept of minimum distance. We evaluate our approach on well-known benchmark datasets and results obtain…
University of Leicester（莱斯特大学）；Lobachevsky University（罗巴切夫斯基大学）；Saint-Petersburg State Electrotechnical University（圣彼得堡国立电工大学）
Artificial Intelligence (AI) systems sometimes make errors and will make errors in the future, from time to time. These errors are usually unexpected, and can lead to dramatic consequences. Intensive development of AI and its practical applications makes the problem of errors more important. Total re-engineering of the systems can create new errors and is not always possible due to the resources involved. The important challenge is to develop fast methods to correct errors without damaging existing skills. We formulated the technical requirements to the 'ideal' correctors. Such correctors incl…
The support vector machine (SVM) is a powerful and widely used classification algorithm. This paper uses the Karush-Kuhn-Tucker conditions to provide rigorous mathematical proof for new insights into the behavior of SVM. These insights provide perhaps unexpected relationships between SVM and two other linear classifiers: the mean difference and the maximal data piling direction. For example, we show that in many cases SVM can be viewed as a cropped version of these classifiers. By carefully exploring these connections we show how SVM tuning behavior is affected by characteristics including: ba…
In this paper we propose a data augmentation method for time series with irregular sampling, Time-Conditional Generative Adversarial Network (T-CGAN). Our approach is based on Conditional Generative Adversarial Networks (CGAN), where the generative step is implemented by a deconvolutional NN and the discriminative step by a convolutional NN. Both the generator and the discriminator are conditioned on the sampling timestamps, to learn the hidden relationship between data and timestamps, and consequently to generate new time series. We evaluate our model with synthetic and real-world datasets. F…
In the last decade, traditional dictionary learning methods have been successfully applied to various pattern classification tasks. Although these methods produce sparse representations of signals which are robust against distortions and missing data, such representations quite often turn out to be unsuitable if the final objective is signal classification. In order to overcome or at least to attenuate such a weakness, several new methods which incorporate discriminative information into sparse-inducing models have emerged in recent years. In particular, methods for discriminative dictionary l…
University of Oslo（奥斯陆大学）；National University of Singapore（新加坡国立大学）
Supervised machine learning applications in the health domain often face the problem of insufficient training datasets. The quantity of labelled data is small due to privacy concerns and the cost of data acquisition and labelling by a medical expert. Furthermore, it is quite common that collected data are unbalanced and getting enough data to personalize models for individuals is very expensive or even infeasible. This paper addresses these problems by (1) designing a recurrent Generative Adversarial Network to generate realistic synthetic data and to augment the original dataset, (2) enabling…
Two proteins are homologous if they have a common evolutionary origin, and the binary classification problem is to identify proteins in a candidate set that are homologous to a particular native protein. The feature (explanatory) variables available for classification are various measures of similarity of proteins. There are multiple classification problems of this type for different native proteins and their respective candidate sets. Homologous proteins are rare in a single candidate set, giving a highly unbalanced two-class problem. The goal is to rank proteins in a candidate set according …
The area under the ROC curve is widely used as a measure of performance of classification rules. However, it has recently been shown that the measure is fundamentally incoherent, in the sense that it treats the relative severities of misclassifications differently when different classifiers are used. To overcome this, Hand (2009) proposed the $H$ measure, which allows a given researcher to fix the distribution of relative severities to a classifier-independent setting on a given problem. This note extends the discussion, and proposes a modified standard distribution for the $H$ measure, which …
We study the effect of imperfect training data labels on the performance of classification methods. In a general setting, where the probability that an observation in the training dataset is mislabelled may depend on both the feature vector and the true label, we bound the excess risk of an arbitrary classifier trained with imperfect labels in terms of its excess risk for predicting a noisy label. This reveals conditions under which a classifier trained with imperfect labels remains consistent for classifying uncorrupted test data points. Furthermore, under stronger conditions, we derive detai…
We study the supervised clustering problem under the two-component anisotropic Gaussian mixture model in high dimensions and in the non-asymptotic setting. We first derive a lower and a matching upper bound for the minimax risk of clustering in this framework. We also show that in the high-dimensional regime, the linear discriminant analysis (LDA) classifier turns out to be sub-optimal in the minimax sense. Next, we characterize precisely the risk of $\ell_2$-regularized supervised least squares classifiers. We deduce the fact that the interpolating solution may outperform the regularized clas…
The Neyman-Pearson (NP) paradigm in binary classification seeks classifiers that achieve a minimal type II error while enforcing the prioritized type I error controlled under some user-specified level $α$. This paradigm serves naturally in applications such as severe disease diagnosis and spam detection, where people have clear priorities among the two error types. Recently, Tong, Feng and Li (2018) proposed a nonparametric umbrella algorithm that adapts all scoring-type classification methods (e.g., logistic regression, support vector machines, random forest) to respect the given type I error…
