Linear dimensionality reduction by maximizing the Chernoff distance in the transformed space

In recent years, the data tends to be very large and complex and it becomes very difficult and tedious to work with large datasets containing huge number of features. That's where Dimensionality Reduction comes into play. Dimensionality Reduction is a pre-processing step in various fields such as machine learning, data mining, statistics etc. and is effective in removing irrelevant and highly redundant data. In this paper, the author's performed a vast literature survey and aims to provide an adequate application based understanding of various dimensionality reduction techniques and to work as a guide to choose right approach of Dimensionality Reduction for better performance in different applications. Here, the authors have also performed detailed experiments on two different datasets for comparative analysis between various linear and non-linear dimensionality reduction techniques to figure out the effectiveness of the techniques used. PCA, a linear dimensionality reduction technique, outperformed all other techniques used in the experiments. In fact, almost all the linear dimensionality reduction techniques outperformed the non-linear techniques on both datasets by a huge error percentage margin.

A theoretical analysis for comparing two linear dimensionality reduction (LDR) techniques, namely Fisher's discriminant (FD) and Loog-Duin (LD) dimensionality reduciton, is presented. The necessary and sufficient conditions for which FD and LD provide the same linear transformation are discussed and proved. To derive these conditions, it is first shown that the two criteria preserve the same maximum value after a diagonalization process is applied, and then the necessary and sufficient conditions for various cases, including coincident covariance matrices, coincident prior probabilities, and for when one of the covariances is the identity matrix. A measure for comparing the two criteria is derived from the necessary and sufficient conditions, and used to empirically show that the conditions are statistically related to the classification error for a post-processing quadratic classifier and the Chernoff distance in the transformed space.

A new linear dimensionality reduction (LDR) technique for pattern classification and machine learning is presented, which, though linear, aims at maximizing the Chernoff distance in the transformed space. The corresponding two-class criterion, which is maximized via a gradient-based algorithm, is presented and initialization procedures are also discussed. Empirical results of this and traditional LDR approaches combined with two well-known classifiers, linear and quadratic, on synthetic and real-life data show that the proposed criterion outperforms the traditional schemes.

For purposes of dimensionality reduction in classification Linear Discriminant Analysis (LDA) is probably the most common approach. In fact, LDA is a linear dimension reduction technique that also returns a classification rule. In the case of heteroscedasticity of the classes, Quadratic Discriminant Analysis (QDA) can be used to determine an appropriate classification rule, but QDA does not serve for dimensionality reduction. Sliced Average Variance Estimation (SAVE) has been shown to be adequate in such situations as implemented in R in the package dr. This paper presents an alternative approach for linear dimensionality reduction for situations of heteroscedastic intraclass covariances, namely Heteroscedastic Discriminant Analysis (HDA) as well as its R implementation. Furthermore, tests are suggested in order to determine the dimension for the discriminative data subspace and a generalization of HDA by regularization of the covariance matrix estimates is proposed. Examples for application of HDA in R are demonstrated as well as a small simulation study turning out that HDA is preferable to SAVE in a situation where the classes differ in both means and covariances.

Embedding the original high dimensional data in a low dimensional space helps to overcome the curse of dimensionality and removes noise. The aim of this work is to evaluate the performance of three different linear dimensionality reduction techniques (DR) techniques namely principal component analysis (PCA), multi dimensional scaling (MDS) and linear discriminant analysis (LDA) on classification of cardiac arrhythmias using probabilistic neural network classifier (PNN). The design phase of classification model comprises of the following stages: preprocessing of the cardiac signal by eliminating detail coefficients that contain noise, feature extraction through daubechies wavelet transform, dimensionality reduction through linear DR techniques specified, and arrhythmia classification using PNN. Linear dimensionality reduction techniques have simple geometric representations and simple computational properties. Entire MIT-BIH arrhythmia database is used for experimentation. The experimental results demonstrates that combination of PNN classifier (spread parameter, σ = 0.08) and PCA DR technique exhibits highest sensitivity and F score of 78.84% and 78.82% respectively with a minimum of 8 dimensions.

Comparison of dimension reduction techniques applied to the analysis of airborne radionuclide activity concentration

Abstract. Linear dimensionality reduction techniques, notably principal component analysis, are widely used in climate data analysis as a means to aid in the interpretation of datasets of high dimensionality. These linear methods may not be appropriate for the analysis of data arising from nonlinear processes occurring in the climate system. Numerous techniques for nonlinear dimensionality reduction have been developed recently that may provide a potentially useful tool for the identification of low-dimensional manifolds in climate data sets arising from nonlinear dynamics. Here, we apply Isomap, one such technique, to the study of El Niño/Southern Oscillation variability in tropical Pacific sea surface temperatures, comparing observational data with simulations from a number of current coupled atmosphere-ocean general circulation models. We use Isomap to examine El Niño variability in the different datasets and assess the suitability of the Isomap approach for climate data analysis. We conclude that, for the application presented here, analysis using Isomap does not provide additional information beyond that already provided by principal component analysis.

Low-rank matrix factorization problems such as non negative matrix factorization (NMF) can be categorized as a clustering or dimension reduction technique. The latter denotes techniques designed to find representations of some high dimensional dataset in a lower dimensional manifold without a significant loss of information. If such a representation exists, the features ought to contain the most relevant features of the dataset. Many linear dimensionality reduction techniques can be formulated as a matrix factorization. In this paper, we combine the conjugate gradient (CG) method with the Barzilai and Borwein (BB) gradient method, and propose a BB scaling CG method for NMF problems. The new method does not require to compute and store matrices associated with Hessian of the objective functions. Moreover, adopting a suitable BB step size along with a proper nonmonotone strategy which comes by the size of convex parameter ηk, results in a new algorithm that can significantly improve the CPU time, efficiency, the number of function evaluation. Convergence result is established and numerical comparisons of methods on both synthetic and real-world datasets show that the proposed method is efficient in comparison with existing methods and demonstrate the superiority of our algorithms.

Analysis of population genetic variation and structure is a common practice for genome-wide studies, including association mapping, ecology, and evolution studies in several crop species. In this study, machine learning (ML) clustering methods, K-means (KM), and hierarchical clustering (HC), in combination with non-linear and linear dimensionality reduction techniques, deep autoencoder (DeepAE) and principal component analysis (PCA), were used to infer population structure and individual assignment of maize inbred lines, i.e., dent field corn (n = 97) and popcorn (n = 86). The results revealed that the HC method in combination with DeepAE-based data preprocessing (DeepAE-HC) was the most effective method to assign individuals to clusters (with 96% of correct individual assignments), whereas DeepAE-KM, PCA-HC, and PCA-KM were assigned correctly 92, 89, and 81% of the lines, respectively. These findings were consistent with both Silhouette Coefficient (SC) and Davies–Bouldin validation indexes. Notably, DeepAE-HC also had better accuracy than the Bayesian clustering method implemented in InStruct. The results of this study showed that deep learning (DL)-based dimensional reduction combined with ML clustering methods is a useful tool to determine genetically differentiated groups and to assign individuals into subpopulations in genome-wide studies without having to consider previous genetic assumptions.

Face recognition is an area of computer vision that has attracted a lot of interest from the research community. A growing demand for robust face recognition software in security applications has driven the development of interesting approaches in this field. A large quantity of research in face recognition deals with visible face images. In the visible spectrum the illumination and face expressions changes represent a significant challenge for the recognition system. To avoid these problems, researchers proposed recently the use of 3D and infrared imaging for face recognition. In this work, we introduce a new framework for infrared face recognition using texture descriptors. This framework exploits linear and non linear dimensionality reduction techniques for face learning and recognition in the texture space. Active and passive infrared imaging modalities are used and comparison with visible face recognition is performed. Two multispectral face recognition databases were used in our experiments: Equinox Database (Visible, SWIR, MWIR, LWIR) and Laval University Multispectral Database (Visible, NIR, MWIR, LWIR). The obtained results show high increase in recognition performance when texture descriptors like LBP (Local Binary Pattern) and LTP (Local Ternary Pattern) are used. The best result was obtained in the short wave infrared spectrum (SWIR) using non linear dimensionality reduction techniques.

Multi-class pairwise linear dimensionality reduction using heteroscedastic schemes

As a widely used linear dimensionality reduction technique, Locality Preserving Projections (LPP) preserves the neighborhood structure of the dataset by finding the optimal linear approximations to the eigenfunctions of the Laplace-Beltrami operator on the manifold, which makes it have several advantages of both linear and nonlinear methods. However, its neighborhood graph is generated by adopting the Euclidean distance as the similarity metric of different samples which leads to the unsatisfying effectiveness of LPP. To address the limitation of Euclidean distance we propose an improved LPP called Manifold Ranking-based LPP (MRLPP) which can effectively preserve the neighborhood structure of the dataset, either globular or non-globular. Experimental results on several datasets demonstrate the effectiveness of our method.Keywordslocality preserving projectionsmanifold rankingneighborhood graph

Reducing dimensionality is a key preprocessing step in many data analysis applications to address the negative effects of the curse of dimensionality and collinearity on model performance and computational complexity, to denoise the data or to reduce storage requirements. Moreover, in many applications it is desirable to reduce the input dimensions by choosing a subset of variables that best represents the entire set without any a priori information available. Unsupervised variable selection techniques provide a solution to this second problem. An autoencoder, if properly regularized, can solve both unsupervised dimensionality reduction and variable selection, but the training of large neural networks can be prohibitive in time sensitive applications. We present an approach called Recovery of Linear Components (RLC), which serves as a middle ground between linear and non-linear dimensionality reduction techniques, reducing autoencoder training times while enhancing performance over purely linear techniques. With the aid of synthetic and real world case studies, we show that the RLC, when compared with an autoencoder of similar complexity, shows higher accuracy, similar robustness to overfitting, and faster training times. Additionally, at the cost of a relatively small increase in computational complexity, RLC is shown to outperform the current state-of-the-art for a semiconductor manufacturing wafer measurement site optimization application.

In recent years, machine learning algorithms have been developing rapidly, becoming increasingly powerful tools in decoding physiological and neural signals. The aim of this dissertation is to develop computational tools, and especially machine learning techniques, to identify the most effective methods for feature extraction and classification of these signals. This is particularly challenging due to the highly non-linear, non-stationery, and artifact- and noise-prone nature of these signals. Among basic human-control tasks, reaching and grasping are ubiquitous in everyday life. I investigated different linear and non-linear dimensionality reduction techniques for feature extraction and classification of electromyography (EMG) during a reach-grasp-lift task. The results highlighted the advantages of completely automated feature-learning by Laplacian Eigenmaps over manual feature engineering, especially when combined with classification, to achieve high accuracy with few training samples. The ability to decode and reduce the complexity of EMG could enable new practical applications for EMG in basic science and in the clinic. It could also help design humanoid and other robots. Beyond EMG, a key objective of my dissertation was to decode brain activity during the decision-making processes that lead to voluntary action. This was based on electroencephalography (EEG) and holds the promise for improved brain-computer interfaces (BCIs), particularly related to motor imagery (MI). We developed an end-to-end convolutional neural network with attentional mechanism together with different data augmentation techniques on two benchmark MI datasets. I also collected a new dataset, recorded using high-density EEG, and containing both MI and motor execution (ME) tasks. This enabled us to directly compare the decodability of MI and ME, investigate optimal channel configurations, and much more. In particular, this facilitates the analysis and decoding of MI on the fly—online and in in real time. Another potential use of EEG is measuring brain activity of people who are floating in body-temperature water in a sensory-deprivation-tank float pod. We compared lying in the float pod versus lying in bed (a control condition). And we found differences between the two, especially in the gamma band. More research is required to understand what these findings mean for levels of stress in the float pod.

Quantile regression (QR) is a well-established method of tail analysis. Application of QR can become very challenging when dealing with high-dimensional data, thus requiring dimension reduction techniques. While the current literature on these techniques focuses on extracting linear combinations of the predictor variables that contain all the information about the conditional quantile, non-linear features can potentially achieve greater dimension reduction. We, therefore, present the first application of transformed dimension reduction for conditional quantiles, which serves as an intermediate step between linear and nonlinear dimension reduction. The idea is to transform the predictors monotonically and then look for low-dimensional projections by applying linear dimension reduction techniques. The performance of the proposed methodology is demonstrated through simulation examples and a real data application.